Permutations & Combinations Questions with Solutions

63 questions29 easy29 medium5 hard

Permutations count arrangements where order matters, while combinations count selections where order does not matter. This topic covers the fundamental principle of counting, factorials, nPr and nCr, arrangements of letters in a word (including repeated letters), circular arrangements, and selecting committees or teams with conditions. The hardest part is deciding whether a question needs a permutation or a combination. Ask yourself whether swapping two chosen items gives a new outcome. If it does, order matters. Conditions such as "two people must sit together" are handled by treating them as a single unit, and "at least one" questions are usually easiest to solve as total minus none. Practise the permutation and combination questions below, each with a step-by-step solution showing how to set up the count.

All Questions Easy Medium Hard

Q1.
The number of diagonals in a polygon of $m$ sides is
A. $\frac{1}{3} m (m - 1)$ B. $\frac{1}{2} m (m - 2)$ C. $\frac{1}{2} m (m - 3)$ D. $\frac{1}{4} m (m - 3)$
easyAsked in MAH MCA CET 2023View solution ›
Q2.
A five-digit number divisible by 3 is to be made using the digits 0, 1, 2, 3, 4, 5 without repetition. Find the number of such numbers.
A. 216B. 260C. 316D. 200
hardAsked in MAH MCA CET 2023View solution ›
Q3.
Let $n$ be the number of injective functions $f : 1, 2, 3, 4 \to 1, 2, 3, 4, 5, 6, 7, 8$ sending an even number to an even number. If $n = 2^{a} 3^{b} 5^{c}$ , then $a + b + c$ is:
A. 5B. 6C. 7D. 8
mediumAsked in NIMCET 2026View solution ›
Q4.
Find the total number of ways to put $20$ balls into $5$ boxes so that the first box contains only one ball.
A. $10 \times 3^{10}$ B. $20 \times 4^{19}$ C. $30 \times 2^{9}$ D. $25 \times 7^{5}$
mediumAsked in MAH MCA CET 2025View solution ›
Q5.
Find the number of arrangements of the letters of the word $, I N D E P E N D E N C E,$ such that
A. $1260, 1646400$ B. $10500, 1600000$ C. $15000, 1663200$ D. $11200, 1680000$
mediumAsked in MAH MCA CET 2025View solution ›
Q6.
Find the total number of ways to put $20$ balls into $5$ boxes so that the first box contains only one ball.
A. $10 \times 3^{10}$ B. $20 \times 4^{19}$ C. $30 \times 2^{9}$ D. $25 \times 7^{5}$
mediumAsked in MAH MCA CET 2025View solution ›
Q7.
The number of parallelograms formed by a set of $4$ parallel lines intersecting another set of $3$ parallel lines is
A. 6B. 9C. 12D. 18
easyAsked in MAH MCA CET 2025View solution ›
Q8.
Straight lines are drawn by joining m points on a straight line to n points on another line. Then excluding the given points, the number of point of intersection of the lines drawn is (no two lines drawn are parallel and no three lines are concurrent).
A. $\frac{1}{4} mn (m - 1) (n - 1)$ B. $\frac{1}{2} mn (m - 1) (n - 1)$ C. $\frac{1}{4} m^{2} n^{2}$ D. $\frac{1}{2} m^{2} n^{2}$
hardAsked in MAH MCA CET 2024View solution ›
Q9.
In an examination of nine papers, a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful is
A. 255B. 256C. 128D. $9 \times 8!$
easyAsked in NIMCET 2023View solution ›
Q10.
Let $T_{n}$ denote the number of triangles which can be formed by using the vertices of a regular polygon of n sides. If $T_{n + 1} - T_{n} = 21$ , then n equals
A. 5B. 7C. 6D. 4
easyAsked in NIMCET 2013View solution ›
Q11.
In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?
A. 810B. 1440C. 2880D. 50400
mediumAsked in NIMCET 2013View solution ›
Q12.
The number of non-negative integers less than 1000 that contain the digit 1 is
A. $9^{2}$ B. $9^{3}$ C. $10^{2} - 9^{2}$ D. $10^{3} - 9^{3}$
mediumAsked in NIMCET 2013View solution ›
Q13.
The total number of numbers that can be formed using the digits 3, 5 and 7 only, if no repetitions are allowed, is
A. 39B. 105C. 15D. 27
easyAsked in NIMCET 2013View solution ›
Q14.
The sum $^{20} C_{8} + ^{20} C_{9} + ^{21} C_{10} + ^{22} C_{11} - ^{23} C_{11}$ is
A. $^{22} C_{12}$ B. $^{23} C_{12}$ C. $^{21} C_{10}$ D. 0
mediumAsked in NIMCET 2012View solution ›
Q15.
The number of words that can be formed by using the letters of the word MATHEMATICS that start as well as end with T is
A. 80720B. 90720C. 20860D. 37528
mediumAsked in NIMCET 2012View solution ›
Q16.
The number of different licence plates that can be formed in the format of 3 English letters (A to Z) followed by 4 digits (0 to 9), with repetition allowed in letters and digits, is
A. $26^{3} \times 10^{4}$ B. $26^{3} + 10^{4}$ C. 36D. $26^{3}$
easyAsked in NIMCET 2012View solution ›
Q17.
A polygon has 44 diagonals. The number of its sides is
A. 9B. 10C. 11D. 12
easyAsked in NIMCET 2011 +1View solution ›
Q18.
The number of ways of forming different $9$ -digit numbers by rearranging the digits of $223355888$ so that the odd digits occupy even positions is
A. $16$ B. $36$ C. $60$ D. $180$
mediumAsked in NIMCET 2011View solution ›
Q19.
How many arrangements of the letters of the word DETAIL have the vowels only in the odd positions?
A. $32$ B. $36$ C. $48$ D. $60$
easyAsked in NIMCET 2011View solution ›
Q20.
Steel Express runs between Tatanagar and Howrah and has five stoppages in between. Find the number of different kinds of one-way second class tickets that Indian Railways must print to cover all possible passenger trips.
A. 49B. 42C. 21D. 7
easyAsked in NIMCET 2010View solution ›

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