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Permutations and Combinations

214 Questions Start Comprehensive Algebra Test
All Exams Comprehensive Algebra
Q101 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
Concept: Restricted Permutations and Combinations with/without Repetition
How many numbers greater than 1000, but not greater than 4000 can be formed with the digits 0, 1, 2, 3, 4, if: (i) repetition of digits is allowed? (ii) repetition of digits is not allowed?
A.
$375, 72$
B.
$376, 72$
C.
$376, 96$
D.
$375, 96$
Q102 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
Concept: Divisibility Rule of 3 and Combinatorial Selection
How many five digits numbers divisible by 3 can be formed using the digits 0, 1, 2, 3, 4, 7 and 8 if each digit is to be used at most once.
A.
$720$
B.
$744$
C.
$680$
D.
$756$
Q103 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
Concept: Fundamental Multiplication Principle of Counting
In how many ways can 6 boys and 5 girls be arranged for a group photograph if the girls are to sit on chairs in a row and the boys are to stand in a row behind them?
A.
$86400$
B.
$72000$
C.
$43200$
D.
$51840$
Q104 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
Concept: Forming Numbers with Specific Magnitude Constraints
How many different numbers greater than 5000 can be formed with the digits 0, 1, 5, 9, no digit being repeated?
A.
$24$
B.
$12$
C.
$18$
D.
$16$
Q105 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
Concept: Counting Numbers in Given Range
Find the number of numbers lying between 300 and 4000 that can be formed with the digits 0, 1, 2, 3, 4, 5; no digit being repeated.
A.
$240$
B.
$180$
C.
$210$
D.
$300$
Q106 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
Concept: Parity of Digit Sums and Fundamental Counting Rule
How many five digits numbers are there whose sum of digits is odd?
A.
$45000$
B.
$50000$
C.
$40000$
D.
$90000$
Q107 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
Concept: Divisibility Rules and Position Constraints
If repetition of digits is not allowed how many numbers of four digits divisible by 5 can be formed with the digits 0, 4, 5, 6, 7?
A.
$42$
B.
$48$
C.
$36$
D.
$24$
Q108 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
Concept: Divisibility by 25 without Repetition
Find the number of numbers divisible by 25 that can be formed using only the digits 1, 2, 3, 4, 5 and 0 taken five at a time.
A.
$42$
B.
$36$
C.
$48$
D.
$54$
Q109 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combinations vs Permutations (Selection vs Ordering)
A sales representative has 10 accounts in a certain city.
(a) In how many ways can 3 accounts be selected to call on?
(b) In how many ways can calls be scheduled for 3 of the 10 accounts?
A.
$120$ and $720$
B.
$720$ and $120$
C.
$210$ and $504$
D.
$120$ and $120$
Q110 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combination of 2 Elements from $n$ Distinct Elements
How many matches will be played in a football championship of 16 participants if every two teams play with each other once?
A.
$256$
B.
$120$
C.
$240$
D.
$160$
Q111 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Selection of $r$ Objects from $n$ Objects
Two ambulance attendants are needed in any emergency and there are six men on duty. In how many ways can the two be chosen?
A.
$30$
B.
$12$
C.
$15$
D.
$36$
Q112 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Division into Groups / Selecting Subsets
In how many ways can 7 different books be given to two students if one student is to have 4 books and the other is to have 3 books?
A.
$42$
B.
$35$
C.
$70$
D.
$210$
Q113 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combinatorial Selection and Inclusion Constraints
A teacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once. She finds that she goes to the zoo 84 times more than a particular child goes to the zoo. Find the number of children in her class.
A.
$10$
B.
$12$
C.
$8$
D.
$15$
Q114 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Properties of Combinations and Factorials
Prove that the product of $r$ consecutive integers is divisible by $r!$.
A.
$p = \frac{r!}{n!}$
B.
$\frac{p}{r!} = ^{n+r-1}C_r$ which is an integer
C.
$\frac{p}{r!} = ^nC_r$
D.
$p = r! \times n!$
Q115 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Algebraic Equations Involving Permutations and Combinations
Solve the equation $^{x+3}P_2 = ^{x+2}C_3 + 20$.
A.
$x = 2$
B.
$x = 5$
C.
$x = 3$
D.
$x = 4$
Q116 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Division of Distinct Objects into Specified Group Sizes
Suppose we have 10 social workers and we wish them to work in groups of 2, 3 and 5. How many ways can we split them up into groups of the right size?
A.
$2520$
B.
$1260$
C.
$5040$
D.
$3150$
Q117 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Partitioning a Set into Distinct Named Groups
In this year's school play the cast consists of 5 members of a royal family, 15 soldiers, and 30 peasants. How many ways can the cast of 50 pupils be divided among these three groups?
A.
$^{50}C_5 \times ^{45}C_{15}$
B.
$^{50}C_{30} \times ^{20}C_{15}$
C.
$\frac{50!}{5! 15! 30!}$
D.
All of the above
Q118 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combinations with Repetition and Identical Objects on Restricted Positions
How many ways can 12 identical white pawns and 12 identical black pawns be placed on the black squares of an $8 \times 8$ chessboard?
A.
$^{32}C_{12} \times ^{20}C_{12}$
B.
$^{64}C_{12} \times ^{52}C_{12}$
C.
$^{32}C_{24}$
D.
$^{32}C_{12} \times ^{12}C_{12}$
Q119 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Counting Pairs in Rows and Columns
How many ways are there to place 2 identical rooks in a common row or column of an $8 \times 8$ chessboard?
A.
$16 \times ^8C_2$
B.
$8 \times ^8C_2$
C.
$2 \times 8 \times ^8C_2$
D.
$64 \times ^8C_2$
Q120 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Complementary Counting on Restricted Grids
How many ways are there to place 2 identical kings on an $8 \times 8$ chessboard so that the kings are not in adjacent squares in the same row or column?
A.
$2 \times 8 \times (^8C_2 - 7)$
B.
$16 \times ^8C_2 - 56$
C.
$2 \times 8 \times (^8C_2 - 8)$
D.
$8 \times (^8C_2 - 7)$
Q121 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combinations and Modular Arithmetic / Complementary Counting
All possible two-factor products are formed from the numbers 1, 2, 3, ..., 100. How many numbers out of the total obtained are multiples of 3?
A.
$2211$
B.
$2739$
C.
$4950$
D.
$2310$
Q122 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Fundamental Principle of Counting and Combinations
A basketball coach must select two attackers and two defenders from among three attackers and five defenders. How many different combinations of attackers and defenders can he select?
A.
$30$
B.
$15$
C.
$20$
D.
$25$
Q123 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Addition and Multiplication Principles in Combinatorial Selection
A cricket team of 11 men is to be chosen from 15 men. Four of the 15 can bowl and two others can keep wicket. If the team is must include only one wicket keeper and at least three bowlers, in how many ways can it be chosen?
A.
$456$
B.
$512$
C.
$384$
D.
$420$
Q124 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Case Analysis and Combination Sum Rule
A group of students consists of 4 boys and 5 girls. Find the number of ways of selecting a team of at least 3 boys and 4 girls.
A.
$25$
B.
$30$
C.
$35$
D.
$40$
Q125 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Pascal's Identity for Combinations
Evaluate $^nC_r + \sum_{j=0}^{3} {^{n+j}C_{r+1+j}}$.
A.
$^{n+3}C_{r+3}$
B.
$^{n+4}C_{r+4}$
C.
$^{n+4}C_{r+3}$
D.
$^{n+3}C_{r+4}$
Q126 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Maximum Value of Combination $^{n}C_r$ for Odd $n$
A class has 21 students. The class teacher has been asked to make groups of $r$ students each and go to the zoo taking one group at a time. Find the size of the group for which the teacher goes the maximum number of times to the zoo.
A.
10 or 11
B.
9 or 12
C.
11 or 12
D.
8 or 10
Q127 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Ratio Properties of Combinations and Factorial Simplification
If $^{n+1}C_{r+1} : ^nC_r : ^{n-1}C_{r-1} = 11 : 6 : 3$, then find the value of $nr$.
A.
$30$
B.
$50$
C.
$40$
D.
$60$
Q128 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Consecutive Combination Ratios and System of Linear Equations
If $^{n}C_{r-1} = 36$, $^nC_r = 84$, and $^{n}C_{r+1} = 126$, then find $r$.
A.
$4$
B.
$5$
C.
$3$
D.
$2$
Q129 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Vandermonde's Identity and Combinatorial Proof
Prove or identify the correct algebraic form of Vandermonde's Identity: $^{n+m}C_r = \sum_{k=0}^{r} {^nC_k \cdot ^mC_{r-k}}$.
A.
$^nC_0 \cdot ^mC_r + ^nC_1 \cdot ^mC_{r-1} + \dots + ^nC_r \cdot ^mC_0$
B.
$^nC_r + ^mC_r$
C.
$^nC_r \cdot ^mC_r$
D.
$\sum_{k=0}^{r} {^nC_k \cdot ^mC_k}$
Q130 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combinations and Elementary Probability Principles with Playing Cards
What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these:
(i) four cards are of the same suit,
(ii) four cards belong to four different suits,
(iii) are face cards,
(iv) two are red cards and two are black cards,
(v) cards are of the same color?
A.
Total: $270725$; (i) $2860$, (ii) $13^4$, (iii) $495$, (iv) $105625$, (v) $29900$
B.
Total: $270725$; (i) $2860$, (ii) $13^4$, (iii) $495$, (iv) $29900$, (v) $105625$
C.
Total: $270725$; (i) $1430$, (ii) $13^4$, (iii) $495$, (iv) $105625$, (v) $29900$
D.
Total: $270725$; (i) $2860$, (ii) $52^4$, (iii) $495$, (iv) $105625$, (v) $29900$
Q131 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Permutation (Order Matters) vs Combination (Order Does Not Matter)
Which of the following everyday activities involve permutations, and which involve combinations only?
(i) Choosing a cricket team from a class of 24 boys.
(ii) Deciding which of the boys will bat first, second, and so on.
(iii) Deciding which three books to borrow from a library.
(iv) Planning the best sequence in which to decorate the rooms in my house.
(v) Planting a row of rose-bushes so that red and yellow flowers alternate.
A.
(i) Combination, (ii) Permutation, (iii) Combination, (iv) Permutation, (v) Permutation
B.
(i) Permutation, (ii) Combination, (iii) Combination, (iv) Permutation, (v) Permutation
C.
(i) Combination, (ii) Permutation, (iii) Permutation, (iv) Combination, (v) Combination
D.
(i) Permutation, (ii) Permutation, (iii) Combination, (iv) Combination, (v) Permutation
Q132 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Geometric Combinations from Vertices
How many triangles can be constructed by connecting the vertices of an octagon?
A.
$48$
B.
$56$
C.
$64$
D.
$336$
Q133 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Selection of Objects (Combinations)
How many ways can a traffic warden with 3 parking tickets left in his book, give them to three out of the seven cars he find that have over-parked?
A.
$210$
B.
$42$
C.
$35$
D.
$21$
Q134 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Forming Subsets of Equal Size
A party of eight people arrive at a river and a boatman agrees to take them across in two groups of four. In how many ways can the first boatload be chosen?
A.
$70$
B.
$1680$
C.
$35$
D.
$140$
Q135 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combinatoric Selection and Time Intervals
In a local police station, a different combination of three constables is selected daily for duty. If there are ten constables available and the system operates every day except Sunday, how many weeks will go by before the same three constables are on duty again together?
A.
$10$ weeks
B.
$20$ weeks
C.
$12$ weeks
D.
$15$ weeks
Q136 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combinations and Modular Arithmetic / Complementary Counting
All possible two factor products are formed from the numbers 1, 2, 3, ..., 100. How many numbers out of the total obtained are multiples of 3?
A.
$2211$
B.
$2739$
C.
$4950$
D.
$2310$
Q137 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Domain Constraints of Combinations $^nC_r$
Find the domain of definition of the function $f(x) = ^{2x-8}C_{x+1}$ and the set of its values.
A.
Domain: $\{9, 10\}$, Set of values: $\{10, 66\}$
B.
Domain: $\{9, 10, 11\}$, Set of values: $\{10, 45, 66\}$
C.
Domain: $\{8, 9, 10\}$, Set of values: $\{8, 10, 66\}$
D.
Domain: $\{9, 10\}$, Set of values: $\{10, 45\}$
Q138 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Ratio of Consecutive Combinations
Solve the given inequalities:
(a) $^{10}C_{x-1} > 2 \cdot ^{10}C_x$
(b) $8 \cdot ^{105}C_x < 3 \cdot ^{105}C_{x+1}$
A.
(a) $x \ge 8$, (b) $x < 28$
B.
(a) $x \ge 8$, (b) $x > 28$
C.
(a) $x \le 7$, (b) $x < 28$
D.
(a) $x \ge 8$, (b) $x > 27$
Q139 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combinatorial Inequality Solving
Find the positive integral values of $x$ such that $^{x-1}C_4 - ^{x-1}C_3 - \frac{5}{4}(x-2)(x-3) < 0$.
A.
$x \in \{5, 6, 7, 8, 9\}$
B.
$x \in \{5, 6, 7, 8\}$
C.
$x \in \{1, 2, 3, 4, 5\}$
D.
$x \in \{5, 6, 7\}$
Q140 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Multinomial Partitioning into Unassigned Groups
Twelve people are waiting for a bus. When it comes it only has room for 6 on top and 4 downstairs. So in how many ways could you form the three groups (those who go upstairs, downstairs or wait)?
A.
$138600$
B.
$277200$
C.
$11550$
D.
$13860$
Q141 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Combinatorial Equations in Game Tournaments
There are $m$ men and 2 women participating in a chess tournament. Every participant plays two games with every other participant. If the number of games played by the men between themselves exceeds by 66 the number of games played between men and women, then find the value of $m$ and the total number of games played in the tournament.
A.
$m = 11$, total games = $156$
B.
$m = 12$, total games = $182$
C.
$m = 10$, total games = $132$
D.
$m = 9$, total games = $110$
Q142 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Selection of Objects from Distinct Subsets
A bag contains 5 black and 6 red balls, all balls being different. Determine the number of ways in which 2 black and 3 red balls can be selected.
A.
$200$
B.
$150$
C.
$100$
D.
$250$
Q143 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Sequential Selection with Permutation and Combination
A meeting of 40 people must choose a chairman, a secretary and 5 members of a committee. How many different committees can be formed?
A.
$^{40}P_2 \times ^{38}C_5$
B.
$^{40}C_7 \times 7$
C.
$^{40}C_2 \times ^{38}C_5$
D.
$^{40}P_7$
Q144 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Independent Combinatorial Choices
A team consists of two house painters, three plasterers and one joiner. How many different teams can be formed from a staff of fifteen house painters, ten plasterers and five joiners?
A.
$63000$
B.
$31500$
C.
$126000$
D.
$15750$
Q145 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Complementary Counting for "At Least One"
Ten cards have been taken from a pack of 52. In how many cases will there be at least one ace among the selected cards?
A.
$^{52}C_{10} - ^{48}C_{10}$
B.
$^{48}C_{10}$
C.
$^{52}C_{10} - ^4C_1 \times ^{48}C_9$
D.
$^4C_1 \times ^{48}C_9$
Q146 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Selection with Dependent Suit Restrictions
How many ways are there of choosing a hand of 6 cards containing an ace and a king of the same suit from a pack of 52 cards?
A.
$4 \times ^{50}C_4$
B.
$4 \times ^{48}C_4$
C.
$^{52}C_6 - ^{48}C_6$
D.
$13 \times ^{48}C_4$
Q147 Comprehensive Algebra Combinations MCQ
26 Jul 2026
Concept: Constrained Category Selection
In how many ways can a cricket team be selected from a group of 25 players containing 10 batsmen, 8 bowlers, 5 all-rounders and 2 wicket keepers? Assume that the team of 11 players requires 5 batsmen, 3 all-rounders, 2 bowlers and 1 wicket keeper.
A.
$141120$
B.
$70560$
C.
$282240$
D.
$142000$
Q148 Comprehensive Algebra Permutations of Alike objects MCQ
26 Jul 2026
Concept: Permutations of Objects Not All Distinct (Alike Objects)
In how many ways can the letters of the word DADDY be permuted among themselves?
A.
60
B.
120
C.
20
D.
24
Q149 Comprehensive Algebra Permutations of Alike objects MCQ
26 Jul 2026
Concept: Permutations of Objects Not All Distinct (Alike Objects)
In how many ways can you permute the letters of the word VIVEKANANDA?
A.
$1663200$
B.
$3326400$
C.
$831600$
D.
$4989600$
Q150 Comprehensive Algebra Permutations of Alike objects MCQ
26 Jul 2026
Concept: Permutations of Multiset / Like Elements
A train time table must be compiled for various days of the week so that two trains a day depart for three days, one train a day for two days, and three trains a day for two days. How many different time tables can be compiled?
A.
$420$
B.
$210$
C.
$105$
D.
$630$