Question 1: Palette 1/24

JEE Advanced 2022 MCQ

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen ?

A

21816

B

85536

C

12096

D

156816

Question 2: Palette 2/24

JEE Advanced 2021 MSQ

Let

${S_1} = \left\{ {(i,j,k):i,j,k \in \{ 1,2,....,10\} } \right\}$,

${S_2} = \left\{ {(i,j):1 \le i < j + 2 \le 10,i,j \in \{ 1,2,...,10\} } \right\}$,

${S_3} = \left\{ {(i,j,k,l):1 \le i < j < k < l,i,j,k,l \in \{ 1,2,...,10\} } \right\}$ and

${S_4} = \{ (i,j,k,l):i,j,k$ and $l$ are distinct elements in {1, 2, ...., 10}.

If the total number of elements in the set S_r is n_r, r = 1, 2, 3, 4, then which of the following statements is(are) TRUE?

A

n₁ = 1000

B

n₂ = 44

C

n₃ = 220

D

${{{n_4}} \over {12}} = 420$

Question 3: Palette 3/24

JEE Advanced 2018 MCQ

In a high school, a committee has to be formed from a group of 6 boys M₁, M₂, M₃, M₄, M₅, M₆ and 5 girls G₁, G₂, G₃, G₄, G₅.

(i) Let $\alpha $₁ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.

(ii) Let $\alpha $₂ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.

i) Let $\alpha $₃ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.

(iv) Let $\alpha $₄ be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both M₁ and G₁ are NOT in the committee together.

LIST-I	LIST-II
P. The value of $\alpha_1$ is	1. 136
Q. The value of $\alpha_2$ is	2. 189
R. The value of $\alpha_3$ is	3. 192
S. The value of $\alpha_4$ is	4. 200
	5. 381
	6. 461

The correct option is

A

P $ \to $ 4; Q $ \to $ 6; R $ \to $ 2; S $ \to $ 1

B

P $ \to $ 1; Q $ \to $ 4; R $ \to $ 2; S $ \to $ 3

C

P $ \to $ 4; Q $ \to $ 6; R $ \to $ 5; S $ \to $ 2

D

P $ \to $ 4; Q $ \to $ 2; R $ \to $ 3; S $ \to $ 1

Question 4: Palette 4/24

JEE Advanced 2016 MCQ

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be select from this club including the selection of a captain (from among these 4 members ) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

A

380

B

320

C

260

D

95

Question 5: Palette 5/24

JEE Advanced 2014 MCQ

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

A

264

B

265

C

53

D

67

Question 6: Palette 6/24

JEE Advanced 2012 MCQ

Let ${{a_n}}$ denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Let ${{b_n}}$ = the number of such n-digit integers ending with digit 1 and ${{c_n}}$ =the number of such n-digit integers ending with digit 0.

The value of ${{b_6}}$ is

A

7

B

8

C

9

D

11

Question 7: Palette 7/24

JEE Advanced 2012 MCQ

Let ${{a_n}}$ denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Let ${{b_n}}$ = the number of such n-digit integers ending with digit 1 and ${{c_n}}$ =the number of such n-digit integers ending with digit 0.

Which of the following is correct?

A

${a_{17}} = {a_{16}} + {a_{15}}$

B

${c_{17}} \ne {c_{16}} + {c_{15}}$

C

${b_{17}} \ne {b_{16}} + {c_{16}}$

D

${a_{17}} = {c_{17}} + {b_{16}}$

Question 8: Palette 8/24

JEE Advanced 2012 MCQ

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

A

75

B

150

C

210

D

243

Question 9: Palette 9/24

JEE Advanced 2009 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

55

B

66

C

77

D

88

Question 10: Palette 10/24

JEE Advanced 2008 MCQ

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

	Column I		Column II
(A)	The number of permutations containing the word ENDEA is	(P)	5!
(B)	The number of permutations in which the letter E occurs in the first and the last position is	(Q)	2 $\times$ 5!
(C)	The number of permutations in which none of the letters D, L, N occurs in the last five positions is	(R)	7 $\times$ 5!
(D)	The number of permutations in which the letters A, E, O occur only in odd positions is	(S)	21 $\times$ 5!

A

(A) - p ; (B) - s; (C) - q ; (D) - q

B

(A) - q ; (B) - q ; (C) - s ; (D) - p

C

(A) - p ; (B) - s; (C) - p ; (D) - r

D

(A) - p ; (B) - r ; (C) - q ; (D) - p

Question 11: Palette 11/24

JEE Advanced 2007 MCQ

The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

A

360

B

192

C

96

D

48

Question 12: Palette 12/24

JEE Advanced 2007 MCQ

The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is :

A

360

B

192

C

96

D

48

Question 13: Palette 13/24

JEE Advanced 2005 MCQ

A rectangle with sides of lenght (2m - 1) and (2n - 1) units is divided into squares of unit lenght by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is

A

${(m + n - 1)^2}$

B

${4^{m + n - 1}}$

C

${m^2}\,{n^2}$

D

$m(m + 1)\,n\,(m + 1)$

Question 14: Palette 14/24

JEE Advanced 2005 MCQ

If the LCM of p, q is ${r^2}\,{r^4}\,{s^2}$, where r, s, t are prime numbers and p, q are the positive integers then number of ordered pair (p, q) is

A

252

B

254

C

225

D

224

Question 15: Palette 15/24

JEE Advanced 2002 MCQ

The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently is

A

40

B

60

C

80

D

100

Question 16: Palette 16/24

JEE Advanced 2001 MCQ

Let ${T_n}$ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If ${T_{n + 1}} - {T_n} = 21$, then n equals

A

5

B

7

C

6

D

4

Question 17: Palette 17/24

JEE Advanced 2000 MCQ

How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?

A

16

B

36

C

60

D

180

Question 18: Palette 18/24

JEE Advanced 1998 MCQ

An n-digit number is a positive number with exactly digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

A

6

B

7

C

8

D

9

Question 19: Palette 19/24

JEE Advanced 1989 MCQ

A five-digit numbers divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5, without repetition. The total number of ways this can be done is

A

216

B

240

C

600

D

3125

Question 20: Palette 20/24

JEE Advanced 1985 MCQ

The product of any r consecutive natural numbers is always divisible by r!

A

TRUE

B

FALSE

Question 21: Palette 21/24

JEE Advanced 1982 MCQ

Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4; and then the men select the chairs from amongst the remaining. The number of possible arrangements is

A

${}^6{C_3} \times {}^4{C_2}$

B

${}^4{P_2} \times {}^4{P_3}$

C

${}^4{C_2} \times {}^4{P_3}$

D

none of these

Question 22: Palette 22/24

JEE Advanced 1982 MCQ

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the numbers of words which have at least one letter repeated are

A

69760

B

30240

C

99748

D

none of these

Question 23: Palette 23/24

JEE Advanced 1982 MCQ

The value of the expression $\,{}^{47}{C_4} + \sum\limits_{j = 1}^5 {^{52 - j}\,{C_3}} $ is equal to

A

${}^{47}{C_5}$

B

${}^{52}{C_5}$

C

${}^{52}{C_4}$

D

none of these

Question 24: Palette 24/24

JEE Advanced 1979 MCQ

${}^n{C_{r - 1}} = 36,{}^n{C_r} = 84\,\,and\,\,{}^n{C_{r + 1}} = 126$, then r is :

A

1

B

2

C

3

D

None of these.