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.
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 ›
Q3.
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 ›
Q4.
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 ›
Q5.
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 ›
Q6.
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 ›
Q7.
A polygon has 44 diagonals. The number of its sides is
A. 9B. 10C. 11D. 12
easyAsked in NIMCET 2011 +1View solution ›
Q8.
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 ›
Q9.
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 ›
Q10.
How many different paths in the $x y$ -plane are there from $(1, 3)$ to $(5, 6)$ , if a path proceeds one step at a time either right (R) or upward (U)?
A. $35$ B. $40$ C. $45$ D. None of these
easyAsked in NIMCET 2009View solution ›
Q11.
There are 10 points in a plane, out of which 6 are collinear (no other three are collinear). The number of triangles formed by joining these points is:
A. $100$ B. $120$ C. $150$ D. None of these
easyAsked in NIMCET 2009View solution ›
Q12.
In an objective type examination, $120$ objective type questions are there, each with $4$ options P, Q, R and S. A candidate can choose either one of these options or can leave the question unanswered. How many different ways exist for answering this question paper?
A. $5^{120}$ B. $4^{120}$ C. $120^{5}$ D. $120^{4}$
easyAsked in NIMCET 2008View solution ›
Q13.
$12$ members were present at a board meeting. Each member shook hands with all of the other members before and after the meeting. How many handshakes were there?
A. $118$ B. $127$ C. $132$ D. $264$
easyAsked in NIMCET 2008View solution ›
Q14.
How many words can be formed starting with letter D taking all letters from the word DELHI so that the letters are not repeated:
A. 4B. 12C. 24D. 20
easyAsked in NIMCET 2020View solution ›
Q15.
Naresh has 10 friends, and he wants to invite 6 of them to a party. How many times will 3 particular friends never attend the party?
A. 8B. 7C. 720D. 35
easyAsked in NIMCET 2020View solution ›
Q16.
If $\frac{n !}{2 ! ( n - 2 ) !}$ and $\frac{n !}{4 ! ( n - 4 ) !}$ are in the ratio $2 : 1$ , then the value of $n$ is
A. 0B. 2C. 4D. 5
easyAsked in NIMCET 2021View solution ›
Q17.
There are 50 questions in a paper. The number of ways in which a student can attempt one or more questions is
A. $2^{50} - 1$ B. $2^{50} + 1$ C. $2^{50} - 2$ D. $2^{50} + 2$
easyAsked in NIMCET 2021View solution ›
Q18.
A password consists of two alphabets from English followed by three numbers chosen from 0 to 3. If repetitions are allowed, the number of different passwords is
A. $^{26} P_{1} \times ^{26} P_{2} \times ^{4} P_{1} \times ^{3} P_{1} \times ^{2} P_{1}$ B. $(^{26} P_{1})^{2} \times (^{4} P_{1})^{3}$ C. $^{26} P_{1} \times ^{26} P_{2} \times ^{4} P_{1} \times ^{4} P_{2} \times ^{4} P_{3}$ D. $(^{26} P_{1} \times ^{4} P_{1})^{2}$
easyAsked in NIMCET 2014View solution ›
Q19.
The number of ways in which 5 days can be chosen in each of the 12 months of a non-leap year, is:
A. $(^{30} C_{5})^{4} \times (^{31} C_{5})^{7} \times ^{28} C_{5}$ B. $(^{30} C_{5})^{6} \times (^{28} C_{5})^{6}$ C. $(^{30} C_{5})^{7} \times (^{31} C_{5})^{4} \times ^{28} C_{5}$ D. $(^{30} C_{5})^{5} \times (^{31} C_{5})^{6} \times ^{28} C_{5}$
easyAsked in NIMCET 2014View solution ›
Q20.
If n and r are integers such that 1 ≤ r ≤ n, then the value of $n \cdot ^{n - 1} C_{r - 1}$ is
A. $^{n} C_{r}$ B. $r (^{n} C_{r})$ C. $n (^{n} C_{r})$ D. $(n - 1) (^{n} C_{r})$
easyAsked in NIMCET 2014View solution ›

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