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.
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 ›
Q2.
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 ›
Q3.
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 ›
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.
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 ›
Q6.
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 ›
Q7.
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 ›
Q8.
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 ›
Q9.
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 ›
Q10.
There are 6 tasks and 6 persons. Task 1 cannot be assigned to person 1 or person 2. Task 2 must be assigned to either person 3 or person 4. Every person is to be given exactly one task. In how many ways can the assignment be done?
A. 144B. 180C. 192D. 360
mediumAsked in NIMCET 2010View solution ›
Q11.
Twelve villages in a district are divided into 3 zones with 4 villages per zone. The telephone department intends to connect the villages so that every two villages in the same zone are connected with three direct lines and every two villages belonging to different zones are connected with two direct lines. How many direct lines are required?
A. 210B. 96C. 54D. 150
mediumAsked in NIMCET 2009View solution ›
Q12.
Using the digits 1, 5, 2, 8 (each exactly once), all possible four-digit numbers are formed. The sum of all such numbers lies between:
A. 10000 & 20000B. 20000 & 50000C. 50000 & 100000D. 10000 & 150000
mediumAsked in NIMCET 2009View solution ›
Q13.
All the letters of the word INDIA are arranged in all possible ways and the words so formed are written in dictionary order. What is the 58th word?
A. NIIDAB. INIDAC. NIDIAD. NIDAI
mediumAsked in NIMCET 2009View solution ›
Q14.
The number of functions $f$ from $A = 0, 1, 2$ into $B = 0, 1, 2, 3, 4, 5, 6, 7$ such that $f (i) \leq f (j)$ for $i < j$ is
A. $^{8} C_{3}$ B. $^{8} C_{3} + 2 (^{8} C_{2})$ C. $^{10} C_{3}$ D. None of these
mediumAsked in NIMCET 2008View solution ›
Q15.
There is a young boy’s birthday party in which 3 friends have attended. The mother has arranged 10 games where a prize is awarded for a winning game. The prizes are identical. If each of the 4 children receives at least one prize, then how many distributions of prizes are possible?
A. 80B. 84C. 70D. 72
mediumAsked in NIMCET 2020View solution ›
Q16.
If $n$ is an integer between 0 and 21, then the value of $n$ for which $n! (21 - n)!$ is minimum is
A. 9B. 10C. 12D. 21
mediumAsked in NIMCET 2021View solution ›
Q17.
The number of ways to arrange the letters of the English alphabet, so that there are exactly 5 letters between a and b, is:
A. $^{24} P_{5}$ B. $^{24} P_{5} \times 20!$ C. $2 \times ^{24} P_{5} \times 20!$ D. $2 \times ^{24} P_{5} \times 24!$
mediumAsked in NIMCET 2014View solution ›
Q18.
How many even integers between 4000 and 7000 have four different digits?
A. 672B. 840C. 504D. 728
mediumAsked in NIMCET 2014View solution ›
Q19.
The number of bit strings of length 8 that start with the bit 0 or end with the bits 11 is
A. 132B. 180C. 256D. 160
mediumAsked in NIMCET 2014View solution ›
Q20.
Let A and B be two sets containing four and two elements respectively. The number of subsets of A × B, each having at least three elements, is
A. 270B. 239C. 219D. 256
mediumAsked in NIMCET 2017View solution ›

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