Question 1: Palette 1/127

JEE Mains 2026 MCQ

Let $\mathrm{S}=\{1,2,3,4,5,6,7,8,9\}$. Let $x$ be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9 -digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

A

$56 x=9 y$

B

$21 x=4 y$

C

$45 x=7 y$

D

$29 x=5 y$

Question 2: Palette 2/127

JEE Mains 2026 MCQ

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is

A

1579

B

1578

C

1580

D

1581

Question 3: Palette 3/127

JEE Mains 2026 MCQ

The largest value of $n$, for which $40^n$ divides $60!$, is

A

14

B

13

C

11

D

12

Question 4: Palette 4/127

JEE Mains 2026 MCQ

The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

A

384

B

403

C

429

D

455

Question 5: Palette 5/127

JEE Mains 2026 MCQ

The largest $n \in \mathbb{N}$, for which $7^n$ divides $101!$, is :

A

18

B

15

C

19

D

16

Question 6: Palette 6/127

JEE Mains 2026 MCQ

The number of strictly increasing functions $f$ from the set $\{1,2,3,4,5,6\}$ to the set $\{1,2,3, \ldots ., 9\}$ such that $f(i) \neq i$ for $1 \leq i \leq 6$, is equal to :

A

21

B

28

C

27

D

22

Question 7: Palette 7/127

JEE Mains 2026 MCQ

A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is :

A

18

B

36

C

39

D

72

Question 8: Palette 8/127

JEE Mains 2026 MCQ

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the $10^{\text {th }}$ floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

A

2184

B

3064

C

7056

D

11340

Question 9: Palette 9/127

JEE Mains 2026 MCQ

The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

A

2670

B

2840

C

2920

D

3600

Question 10: Palette 10/127

JEE Mains 2026 MCQ

A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is :

A

4100

B

4140

C

4230

D

4290

Question 11: Palette 11/127

JEE Mains 2026 MCQ

Let

$\mathrm{A}=\{(a, b, c): a, b, c$ are non-negative integers and $a+b+2 c=22\}$.

Then $n(\mathrm{~A})$ is equal to :

A

121

B

124

C

144

D

169

Question 12: Palette 12/127

JEE Mains 2026 MCQ

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:

A

5040

B

3050

C

3410

D

4320

Question 13: Palette 13/127

JEE Mains 2026 MCQ

Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$-side regular polygon.
If $p_{n+1} - p_n = 66$, then the sum of all distinct prime divisors of $n$ is :

A

7

B

8

C

5

D

6

Question 14: Palette 14/127

JEE Mains 2026 MCQ

The number of seven-digit numbers, that can be formed by using the digits 1, 2, 3, 5 and 7 such that each digit is used at least once, is:

A

15400

B

17800

C

16800

D

29400

Question 15: Palette 15/127

JEE Mains 2026 MCQ

The number of elements in the set $S = \left\{ (r, k) : k \in \mathbb{Z} \text{ and } ^{36}C_{r+1} = \frac{6\left(^{35}C_{r}\right)}{(k^2-3)} \right\}$ is :

A

2

B

4

C

8

D

16

Question 16: Palette 16/127

JEE Mains 2025 MCQ

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is

A

230

B

210

C

200

D

220

Question 17: Palette 17/127

JEE Mains 2025 MCQ

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is

A

145

B

165

C

155

D

135

Question 18: Palette 18/127

JEE Mains 2025 MCQ

Line $L_1$ of slope 2 and line $L_2$ of slope $\frac{1}{2}$ intersect at the origin O . In the first quadrant, $\mathrm{P}_1$, $P_2, \ldots, P_{12}$ are 12 points on line $L_1$ and $Q_1, Q_2, \ldots, Q_9$ are 9 points on line $L_2$. Then the total number of triangles, that can be formed having vertices at three of the 22 points $\mathrm{O}, \mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{12}$, $\mathrm{Q}_1, \mathrm{Q}_2, \ldots, \mathrm{Q}_9$, is:

A

1026

B

1188

C

1134

D

1080

Question 19: Palette 19/127

JEE Mains 2025 MCQ

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :

A

5880

B

840

C

960

D

5760

Question 20: Palette 20/127

JEE Mains 2025 MCQ

The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly five 1 s and exactly three 2 s , is equal to :

A

360

B

2520

C

1820

D

45

Question 21: Palette 21/127

JEE Mains 2025 MCQ

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at 440^th position in this arrangement is :

A

PRNAKU

B

PRKAUN

C

PRKANU

D

PRNAUK

Question 22: Palette 22/127

JEE Mains 2025 MCQ

Let $ P $ be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in $ P $ are formed by using the digits 1, 2 and 3 only, then the number of elements in the set $ P $ is :

A

164

B

158

C

161

D

173

Question 23: Palette 23/127

JEE Mains 2025 MCQ

Let ${ }^n C_{r-1}=28,{ }^n C_r=56$ and ${ }^n C_{r+1}=70$. Let $A(4 \operatorname{cost}, 4 \sin t), B(2 \sin t,-2 \cos t)$ and $C\left(3 r-n, r^2-n-1\right)$ be the vertices of a triangle $A B C$, where $t$ is a parameter. If $(3 x-1)^2+(3 y)^2$ $=\alpha$, is the locus of the centroid of triangle ABC , then $\alpha$ equals

A

18

B

8

C

20

D

6

Question 24: Palette 24/127

JEE Mains 2025 MCQ

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0 , $1,2,3,4,5,6,7$, such that the sum of their first and last digits should not be more than 8 , is

A

5720

B

5719

C

4608

D

4607

Question 25: Palette 25/127

JEE Mains 2025 MCQ

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group $A$ and the remaining 3 from group $B$, is equal to :

A

8925

B

9100

C

8575

D

8750

Question 26: Palette 26/127

JEE Mains 2025 MCQ

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is :

A

34000

B

37000

C

35000

D

36000

Question 27: Palette 27/127

JEE Mains 2025 MCQ

In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is :

A

120

B

96

C

72

D

144

Question 28: Palette 28/127

JEE Mains 2025 MCQ

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

A

6084

B

5148

C

14950

D

4356

Question 29: Palette 29/127

JEE Mains 2024 MCQ

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to:

A

179

B

177

C

175

D

181

Question 30: Palette 30/127

JEE Mains 2024 MCQ

Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f: A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is

A

20

B

120

C

25

D

24

Question 31: Palette 31/127

JEE Mains 2024 MCQ

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{\text {th }}$ position in this arrangement is :

A

NRAPUG

B

NRAGUP

C

NRAPGU

D

NRAGPU

Question 32: Palette 32/127

JEE Mains 2024 MCQ

Let $0 \leq r \leq n$. If ${ }^{n+1} C_{r+1}:{ }^n C_r:{ }^{n-1} C_{r-1}=55: 35: 21$, then $2 n+5 r$ is equal to :

A

62

B

60

C

55

D

50

Question 33: Palette 33/127

JEE Mains 2024 MCQ

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

A

56

B

16

C

24

D

48

Question 34: Palette 34/127

JEE Mains 2024 MCQ

Let the set $S=\{2,4,8,16, \ldots, 512\}$ be partitioned into 3 sets $A, B, C$ with equal number of elements such that $\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}=\mathrm{S}$ and $\mathrm{A} \cap \mathrm{B}=\mathrm{B} \cap \mathrm{C}=\mathrm{A} \cap \mathrm{C}=\phi$. The maximum number of such possible partitions of $S$ is equal to:

A

1640

B

1520

C

1710

D

1680

Question 35: Palette 35/127

JEE Mains 2024 MCQ

60 words can be made using all the letters of the word $\mathrm{BHBJO}$, with or without meaning. If these words are written as in a dictionary, then the $50^{\text {th }}$ word is:

A

OBBJH

B

HBBJO

C

OBBHJ

D

JBBOH

Question 36: Palette 36/127

JEE Mains 2024 MCQ

There are 5 points $P_1, P_2, P_3, P_4, P_5$ on the side $A B$, excluding $A$ and $B$, of a triangle $A B C$. Similarly there are 6 points $\mathrm{P}_6, \mathrm{P}_7, \ldots, \mathrm{P}_{11}$ on the side $\mathrm{BC}$ and 7 points $\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18}$ on the side $\mathrm{CA}$ of the triangle. The number of triangles, that can be formed using the points $\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{18}$ as vertices, is:

A

751

B

776

C

796

D

771

Question 37: Palette 37/127

JEE Mains 2024 MCQ

If $\mathrm{n}$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $\mathrm{n}$ is equal to :

A

47

B

53

C

51

D

43

Question 38: Palette 38/127

JEE Mains 2024 MCQ

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

A

130

B

136

C

142

D

406

Question 39: Palette 39/127

JEE Mains 2024 MCQ

If for some $m, n ;{ }^6 C_m+2\left({ }^6 C_{m+1}\right)+{ }^6 C_{m+2}>{ }^8 C_3$ and ${ }^{n-1} P_3:{ }^n P_4=1: 8$, then ${ }^n P_{m+1}+{ }^{\mathrm{n}+1} C_m$ is equal to

A

380

B

376

C

372

D

384

Question 40: Palette 40/127

JEE Mains 2024 MCQ

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to

A

18

B

16

C

12

D

15

Question 41: Palette 41/127

JEE Mains 2024 MCQ

Let $\alpha=\frac{(4 !) !}{(4 !)^{3 !}}$ and $\beta=\frac{(5 !) !}{(5 !)^{4 !}}$. Then :

A

$\alpha \in \mathbf{N}$ and $\beta \in \mathbf{N}$

B

$\alpha \in \mathbf{N}$ and $\beta \notin \mathbf{N}$

C

$\alpha \notin \mathbf{N}$ and $\beta \in \mathbf{N}$

D

$\alpha \notin \mathbf{N}$ and $\beta \notin \mathbf{N}$

Question 42: Palette 42/127

JEE Mains 2023 MCQ

The total number of three-digit numbers, divisible by 3, which can be formed using the digits $1,3,5,8$, if repetition of digits is allowed, is :

A

21

B

22

C

18

D

20

Question 43: Palette 43/127

JEE Mains 2023 MCQ

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is :

A

324

B

328

C

326

D

327

Question 44: Palette 44/127

JEE Mains 2023 MCQ

The number of five digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits $0,1,3,5,7$ and 9 without repetition, is equal to :

A

132

B

72

C

120

D

96

Question 45: Palette 45/127

JEE Mains 2023 MCQ

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is :

A

103

B

104

C

102

D

101

Question 46: Palette 46/127

JEE Mains 2023 MCQ

The number of triplets $(x, \mathrm{y}, \mathrm{z})$, where $x, \mathrm{y}, \mathrm{z}$ are distinct non negative integers satisfying $x+y+z=15$, is :

A

136

B

80

C

92

D

114

Question 47: Palette 47/127

JEE Mains 2023 MCQ

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is :

A

560

B

1680

C

3360

D

1120

Question 48: Palette 48/127

JEE Mains 2023 MCQ

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which $\mathrm{C}$ and $\mathrm{S}$ do not come together, is $(6 !) \mathrm{k}$, then $\mathrm{k}$ is equal to :

A

5670

B

1890

C

2835

D

945

Question 49: Palette 49/127

JEE Mains 2023 MCQ

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is :

A

16800

B

14800

C

18000

D

33600

Question 50: Palette 50/127

JEE Mains 2023 MCQ

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is :

A

720

B

$7(360)^{2}$

C

$7(720)^{2}$

D

$126(5 !)^{2}$

Question 51: Palette 51/127

JEE Mains 2023 MCQ

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least 3 and at most 6 elements is :

A

782

B

772

C

752

D

792

Question 52: Palette 52/127

JEE Mains 2023 MCQ

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is :

A

578

B

576

C

580

D

582

Question 53: Palette 53/127

JEE Mains 2023 MCQ

The value of $\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}$ is :

A

$\frac{2^{51}}{50 !}$

B

$\frac{2^{51}}{51 !}$

C

$\frac{2^{50}}{50 !}$

D

$\frac{2^{50}}{51 !}$

Question 54: Palette 54/127

JEE Mains 2023 MCQ

The number of ways of selecting two numbers $a$ and $b, a \in\{2,4,6, \ldots ., 100\}$ and $b \in\{1,3,5, \ldots . ., 99\}$ such that 2 is the remainder when $a+b$ is divided by 23 is :

A

186

B

54

C

108

D

268

Question 55: Palette 55/127

JEE Mains 2023 MCQ

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

A

79

B

84

C

89

D

86

Question 56: Palette 56/127

JEE Mains 2023 MCQ

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is :

A

400

B

472

C

507

D

432

Question 57: Palette 57/127

JEE Mains 2023 MCQ

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is :

A

120

B

6

C

72

D

12

Question 58: Palette 58/127

JEE Mains 2023 MCQ

$\sum\limits_{k = 0}^6 {{}^{51 - k}{C_3}} $ is equal to :

A

$\mathrm{{}^{51}{C_4} - {}^{45}{C_4}}$

B

$\mathrm{{}^{51}{C_3} - {}^{45}{C_3}}$

C

$\mathrm{{}^{52}{C_3} - {}^{45}{C_3}}$

D

$\mathrm{{}^{52}{C_4} - {}^{45}{C_4}}$

Question 59: Palette 59/127

JEE Mains 2023 MCQ

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is :

A

48

B

120

C

168

D

220

Question 60: Palette 60/127

JEE Mains 2023 MCQ

The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is :

A

125

B

150

C

225

D

120

Question 61: Palette 61/127

JEE Mains 2022 MCQ

The number of ways to distribute 30 identical candies among four children C₁, C₂, C₃ and C₄ so that C₂ receives at least 4 and at most 7 candies, C₃ receives at least 2 and at most 6 candies, is equal to :

A

205

B

615

C

510

D

430

Question 62: Palette 62/127

JEE Mains 2022 MCQ

The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is :

A

36

B

48

C

60

D

72

Question 63: Palette 63/127

JEE Mains 2021 MCQ

Let P₁, P₂, ......, P₁₅ be 15 points on a circle. The number of distinct triangles formed by points P_i, P_j, P_k such that i +j + k $\ne$ 15, is :

A

12

B

419

C

443

D

455

Question 64: Palette 64/127

JEE Mains 2021 MCQ

If ${}^n{P_r} = {}^n{P_{r + 1}}$ and ${}^n{C_r} = {}^n{C_{r - 1}}$, then the value of r is equal to :

A

1

B

4

C

2

D

3

Question 65: Palette 65/127

JEE Mains 2021 MCQ

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is :

A

26664

B

122664

C

122234

D

22264

Question 66: Palette 66/127

JEE Mains 2021 MCQ

If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :

A

240

B

360

C

333

D

364

Question 67: Palette 67/127

JEE Mains 2021 MCQ

Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

A

5

B

2

C

4

D

6

Question 68: Palette 68/127

JEE Mains 2021 MCQ

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then ($\beta$ $-$ $\alpha$) is equal to :

A

717

B

795

C

1890

D

1173

Question 69: Palette 69/127

JEE Mains 2021 MCQ

A natural number has prime factorization given by n = 2^x3^y5^z, where y and z are such
that y + z = 5 and y^$-$1 + z^$-$1 = ${5 \over 6}$, y > z. Then the number of odd divisions of n, including 1, is :

A

11

B

6

C

12

D

6x

Question 70: Palette 70/127

JEE Mains 2021 MCQ

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is :

A

35

B

42

C

82

D

77

Question 71: Palette 71/127

JEE Mains 2021 MCQ

The total number of positive integral solutions (x, y, z) such that xyz = 24 is :

A

36

B

24

C

45

D

30

Question 72: Palette 72/127

JEE Mains 2021 MCQ

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :

A

1050

B

575

C

560

D

1625

Question 73: Palette 73/127

JEE Mains 2020 MCQ

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?

A

2! 3! 4!

B

(3!)³.(4!)

C

3! (4!)³

D

(3!)².(4!)

Question 74: Palette 74/127

JEE Mains 2020 MCQ

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is :

A

2250

B

2255

C

3000

D

1500

Question 75: Palette 75/127

JEE Mains 2020 MCQ

The value of (2.¹P₀ – 3.²P₁ + 4.³P₂ .... up to 51^th term)
+ (1! – 2! + 3! – ..... up to 51^th term) is equal to :

A

1

B

1 + (51)!

C

1 – 51(51)!

D

1 + (52)!

Question 76: Palette 76/127

JEE Mains 2020 MCQ

Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is :

A

201

B

199

C

101

D

200

Question 77: Palette 77/127

JEE Mains 2020 MCQ

If the number of five digit numbers with distinct digits and 2 at the 10^th place is 336 k, then k is equal to :

A

6

B

8

C

4

D

7

Question 78: Palette 78/127

JEE Mains 2020 MCQ

If a, b and c are the greatest value of ¹⁹C_p, ²⁰C_q and ²¹C_r respectively, then :

A

${a \over {11}} = {b \over {22}} = {c \over {21}}$

B

${a \over {10}} = {b \over {22}} = {c \over {21}}$

C

${a \over {10}} = {b \over {11}} = {c \over {42}}$

D

${a \over {11}} = {b \over {22}} = {c \over {42}}$

Question 79: Palette 79/127

JEE Mains 2020 MCQ

The number of ordered pairs (r, k) for which
6.³⁵C_r = (k² - 3). ³⁶C_{r + 1}, where k is an integer, is :

A

6

B

3

C

2

D

4

Question 80: Palette 80/127

JEE Mains 2020 MCQ

Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is :

A

${5 \over 2}\left( {6!} \right)$

B

${6!}$

C

5⁶

D

${1 \over 2}\left( {6!} \right)$

Question 81: Palette 81/127

JEE Mains 2019 MCQ

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to :

A

24

B

25

C

27

D

28

Question 82: Palette 82/127

JEE Mains 2019 MCQ

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is :

A

2²⁰ - 1

B

2²⁰

C

2²⁰ + 1

D

2²¹

Question 83: Palette 83/127

JEE Mains 2019 MCQ

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is :

A

180

B

210

C

170

D

190

Question 84: Palette 84/127

JEE Mains 2019 MCQ

The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated is :

A

36

B

60

C

72

D

48

Question 85: Palette 85/127

JEE Mains 2019 MCQ

A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the committee is formed with at least 6 males and n is the number of ways the committee is formed with at least 3 females, then :

A

n = m – 8

B

m = n = 78

C

m + n = 68

D

m = n = 68

Question 86: Palette 86/127

JEE Mains 2019 MCQ

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :

A

306

B

288

C

310

D

360

Question 87: Palette 87/127

JEE Mains 2019 MCQ

All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number of such numbers in which the odd digits occupy even places is :

A

175

B

162

C

160

D

180

Question 88: Palette 88/127

JEE Mains 2019 MCQ

There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of m is :

A

12

B

9

C

7

D

11

Question 89: Palette 89/127

JEE Mains 2019 MCQ

Consider three boxes, each containing, 10 balls labelled 1, 2, … , 10. Suppose one ball is randomly drawn from each of the boxes. Denote by n_i, the label of the ball drawn from the i^th box, (i = 1, 2, 3). Then, the number of ways in which the balls can be chosen such that n₁ < n₂ < n₃ is :

A

164

B

240

C

82

D

120

Question 90: Palette 90/127

JEE Mains 2019 MCQ

If  $\sum\limits_{r = 0}^{25} {\left\{ {{}^{50}{C_r}.{}^{50 - r}{C_{25 - r}}} \right\} = K\left( {^{50}{C_{25}}} \right)} ,\,\,$ then K is equal to :

A

2²⁴

B

2²⁵$-$ 1

C

2²⁵

D

(25)²

Question 91: Palette 91/127

JEE Mains 2019 MCQ

Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is :

A

9

B

18

C

36

D

32

Question 92: Palette 92/127

JEE Mains 2019 MCQ

The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7, 9 (repitition of digits allowed) is equal to :

A

374

B

372

C

375

D

250

Question 93: Palette 93/127

JEE Mains 2019 MCQ

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is :

A

500

B

350

C

200

D

300

Question 94: Palette 94/127

JEE Mains 2018 MCQ

The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is not allowed) and are multiple of 3 is :

A

24

B

30

C

36

D

48

Question 95: Palette 95/127

JEE Mains 2018 MCQ

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

A

at least 750 but less than 1000

B

at least 1000

C

less than 500

D

at least 500 but less than 750

Question 96: Palette 96/127

JEE Mains 2018 MCQ

The number of four letter words that can be formed using the letters of the word BARRACK is :

A

120

B

144

C

264

D

270

Question 97: Palette 97/127

JEE Mains 2018 MCQ

n$-$digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is :

A

6

B

7

C

8

D

9

Question 98: Palette 98/127

JEE Mains 2017 MCQ

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy B₁ and a particular girl G₁ never sit adjacent to each other, is :

A

5 $ \times $ 6!

B

6 $ \times $ 6!

C

7!

D

5 $ \times $ 7!

Question 99: Palette 99/127

JEE Mains 2017 MCQ

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is :

A

44^th

B

45^th

C

46^th

D

47^th

Question 100: Palette 100/127

JEE Mains 2017 MCQ

A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is:

A

468

B

469

C

484

D

485

Question 101: Palette 101/127

JEE Mains 2016 MCQ

If    ${{{}^{n + 2}C{}_6} \over {{}^{n - 2}{P_2}}}$ = 11, then n satisfies the equation :

A

n² + 3n − 108 = 0

B

n² + 5n − 84 = 0

C

n² + 2n − 80 = 0

D

n² + n − 110 = 0

Question 102: Palette 102/127

JEE Mains 2016 MCQ

The sum $\sum\limits_{r = 1}^{10} {\left( {{r^2} + 1} \right) \times \left( {r!} \right)} $ is equal to :

A

(11)!

B

10 $ \times $ (11!)

C

101 $ \times $ (10!)

D

11 $ \times $ (11!)

Question 103: Palette 103/127

JEE Mains 2016 MCQ

The value of $\sum\limits_{r = 1}^{15} {{r^2}} \left( {{{{}^{15}{C_r}} \over {{}^{15}{C_{r - 1}}}}} \right)$ is equal to :

A

560

B

680

C

1240

D

1085

Question 104: Palette 104/127

JEE Mains 2016 MCQ

If the four letter words (need not be meaningful ) are to be formed using the letters from the word “MEDITERRANEAN” such that the first letter is R and the fourth letter is E, then the total number of all such words is :

A

${{11!} \over {{{\left( {2!} \right)}^3}}}$

B

110

C

56

D

59

Question 105: Palette 105/127

JEE Mains 2016 MCQ

If all the words (with or without meaning) having five letters,formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL is :

A

${46^{th}}$

B

${59^{th}}$

C

${52^{nd}}$

D

${58^{th}}$

Question 106: Palette 106/127

JEE Mains 2015 MCQ

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:

A

120

B

72

C

216

D

192

Question 107: Palette 107/127

JEE Mains 2013 MCQ

Let ${T_n}$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If ${T_{n + 1}} - {T_n}$ = 10, then the value of n is :

A

7

B

5

C

10

D

8

Question 108: Palette 108/127

JEE Mains 2013 MCQ

Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A $ \times $ B having 3 or more elements is :

A

219

B

211

C

256

D

220

Question 109: Palette 109/127

JEE Mains 2012 MCQ

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is:

A

880

B

629

C

630

D

879

Question 110: Palette 110/127

JEE Mains 2011 MCQ

Statement - 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is emply is ${}^9{C_3}$.
Statement - 2: The number of ways of choosing any 3 places from 9 different places is ${}^9{C_3}$.

A

Statement - 1 is true, Statement - 2 is true, Statement - 2 is not a correct explanation for Statement - 1.

B

Statement - 1 is true, Statement - 2 is false.

C

Statement - 1 is false, Statement - 2 is true.

D

Statement - 1 is true, Statement - 2 is true, Statement - 2 is a correct explanation for Statement - 1.

Question 111: Palette 111/127

JEE Mains 2011 MCQ

These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then:

A

$N \le 100$

B

$100 < N \le 140$

C

$140 < N \le 190\,$

D

$N > 190$

Question 112: Palette 112/127

JEE Mains 2010 MCQ

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

A

36

B

66

C

108

D

3

Question 113: Palette 113/127

JEE Mains 2009 MCQ

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is :

A

at least 500 but less than 750

B

at least 750 but less than 1000

C

at least 1000

D

less than 500

Question 114: Palette 114/127

JEE Mains 2008 MCQ

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

A

$8.{}^6{C_4}.{}^7{C_4}$

B

$6.7.{}^8{C_4}$

C

$6.8.{}^7{C_4}$.

D

$7.{}^6{C_4}.{}^8{C_4}$

Question 115: Palette 115/127

JEE Mains 2008 MCQ

In a shop there are five types of ice-cream available. A child buys six ice-cream.
Statement - 1: The number of different ways the child can buy the six ice-cream is ${}^{10}{C_5}$.
Statement - 2: The number of different ways the child can buy the six ice-cream is equal to the number of different ways of arranging 6 A and 4 B's in a row.

A

Statement - 1 is false, Statement - 2 is true

B

Statement - 1 is true, Statement - 2 is true, Statement - 2 is a correct explanation for Statement - 1

C

Statement - 1 is true, Statement - 2 is true, Statement - 2 is not a correct explanation for Statement - 1

D

Statement - 1 is true, Statement - 2 is false

Question 116: Palette 116/127

JEE Mains 2007 MCQ

The set S = {1, 2, 3, ........., 12} is to be partitioned into three sets A, B, C of equal size. Thus $A \cup B \cup C = S,\,A \cap B = B \cap C = A \cap C = \phi $. The number of ways to partition S is

A

${{12!} \over {{{(4!)}^3}}}\,\,$

B

${{12!} \over {{{(4!)}^4}}}\,\,$

C

${{12!} \over {3!\,\,{{(4!)}^3}}}$

D

${{12!} \over {3!\,\,{{(4!)}^4}}}$

Question 117: Palette 117/127

JEE Mains 2006 MCQ

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote is

A

5040

B

6210

C

385

D

1110

Question 118: Palette 118/127

JEE Mains 2005 MCQ

If the letter of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

A

601

B

600

C

603

D

602

Question 119: Palette 119/127

JEE Mains 2004 MCQ

How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order

A

480

B

240

C

360

D

120

Question 120: Palette 120/127

JEE Mains 2004 MCQ

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

A

${}^8{C_3}$

B

21

C

${3^8}$

D

5

Question 121: Palette 121/127

JEE Mains 2003 MCQ

The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by

A

$7!\, \times 5!\,\,$

B

$6!\, \times 5!$

C

$30!$

D

$5!\, \times 4!$

Question 122: Palette 122/127

JEE Mains 2003 MCQ

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

A

346

B

140

C

196

D

280

Question 123: Palette 123/127

JEE Mains 2003 MCQ

If ${}^n{C_r}$ denotes the number of combination of n things taken r at a time, then the expression $\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}$ equals

A

$\,{}^{n + 1}{C_{r + 1}}$

B

${}^{n + 2}{C_r}$

C

${}^{n + 2}{C_{r + 1}}$

D

$\,{}^{n + 1}{C_r}$

Question 124: Palette 124/127

JEE Mains 2002 MCQ

Five digit number divisible by 3 is formed using 0, 1, 2, 3, 4 and 5 without repetition. Total number of such numbers are :

A

312

B

3125

C

120

D

216

Question 125: Palette 125/127

JEE Mains 2002 MCQ

Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are :

A

216

B

375

C

400

D

720

Question 126: Palette 126/127

JEE Mains 2002 MCQ

The sum of integers from 1 to 100 that are divisible by 2 or 5 is :

A

3000

B

3050

C

3600

D

3250

Question 127: Palette 127/127

JEE Mains 2002 MCQ

Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is :

A

125

B

105

C

374

D

625