Selection & Combination Questions with Solutions — Set 3 (Q21-Q30)

This set contains Selection and Combination Questions with Solutions — Set 3 (Q21-Q30) covering a mix of question types and difficulty levels — from basic to advanced — exactly as asked in real competitive exams.

Solutions are written in a simple, step-by-step notebook style for easy self-study and quick understanding. Each solution is broken down step by step so even the toughest question feels easy. These questions are hand-picked for students preparing for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and all campus placement aptitude tests. International students preparing for GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests will find these equally useful.

✏️ Attempt each question on your own first — then check the solution below.

21. What is the total number of ways of selecting fruits with at least 1 mango from 5 distinct mangoes & 4 distinct oranges?

22. 15 singers were selected to perform in a party, but at a time only 5 singers can perform, then total how many times performance was done if no same group of 5 singers performed twice?

23. 15 singers were selected to perform in a party, but at a time only 5 singers can perform, then how many times a particular singer performed if no same group of 5 singers performed twice?

24. In how many ways 5 number can bee selected from {1, 2, 3, 4, 5, 6, ……….n} such that all the five numbers are cosecutive?

25. In how many ways 5 numbers can be selected from {1, 2, 3, 4, 5, 6, ………. n} such that not all the five numbers are consecutive?

26. If r, s, t are prime numbers & p, q are positive integers such that the LCM of p, q is r²t²s⁴, then the number of ordered pair (p,q) is?

27. In a certain test, aᵢ students gave wrong answer to at least i question where i = 1, 2, 3, ………, k & no student gave more than k wrong answers. What is the total number of wrong answers?

28. In a certain test, 2⁽ᵏ⁻ᶦ⁾ students gave wrong answers to at least i questions, where i = 1, 2, 3, ………., k & no student gave more than k wrong answers. what is the total number of wrong answers given?

29. 30 players from India, Canada & Russia participatedd in world chess tournament. Number of players from india is 3 more than the number of players from canada. In tournament players from same country are not allowed to play against each other & no two player plays against each other more than once. Then find
(i). The maximum number of matches that was played in the tournament
(ii). If number of matches is maximum then what is the ratio of number of players from India & Russia ?

30. A committee of 5 men & 5 women is to be selected from 9 men & 10 women.
(i). In how many ways this can be done
(ii). In how many ways this can be done if a particular woman is always selected
(iii). In how many ways this can be done if a particular woman is always selected and a man is always rejected?

21. What is the total number of ways of selecting fruits with at least 1 mango from 5 distinct mangoes & 4 distinct oranges?
Solution:-

Number of ways of selecting orange = 2⁴ = 16
∴ total number of ways = 31×16
= 496 Answer

22. 15 singers were selected to perform in a party, but at a time only 5 singers can perform, then total how many times performance was done if no same group of 5 singers performed twice?
Solution:-

Number of singers = 15
Number of singers selected at a time = 5
& Since no group of 5 singers performed twice.
∴ out of 15 singers 5 can be selected in ¹⁵C₅ ways.
Hence Answer = ¹⁵C₅

23. 15 singers were selected to perform in a party, but at a time only 5 singers can perform, then how many times a particular singer performed if no same group of 5 singers performed twice?
Solution:-

Since a particular singer is always selected then number of ways of selecting 4 more singers ouht of remaining 14 players is ¹⁴C₄
Hence Answer = ¹⁴C₄

24. In how many ways 5 number can bee selected from {1, 2, 3, 4, 5, 6, ……….n} such that all the five numbers are cosecutive?
Solution:-

25. In how many ways 5 numbers can be selected from {1, 2, 3, 4, 5, 6, ………. n} such that not all the five numbers are consecutive?
Solution:-
Required number of ways
= number of ways of selecting 5 out of n students – number of ways of selecting 5 consecutive out of n students.
= ⁿC₅ – (n-4)
= ⁿC₅ -n + 4      Answer

26. If r, s, t are prime numbers & p, q are positive integers such that the LCM of p, q is r²t²s⁴, then the number of ordered pair (p,q) is?
Solution:-
Since r, s, t are prime numbers.
∴ selection of p & q will be done in following ways:-

27. In a certain test, aᵢ students gave wrong answer to at least i question where i = 1, 2, 3, ………, k & no student gave more than k wrong answers. What is the total number of wrong answers?
Solution:-

Number of students who gave i or more than i wrong answers = aᵢ
Number of students who gave (i+1) or more than (i+1) wrong answers = aᵢ₊₁

∴ Number of students who gave wrong answers to exactly i questions = aᵢ – aᵢ₊₁
∴ total number of wrong answers = \(\sum\limits_{i = 1}^k {i.({a_i} – {a_{i + 1}})} \)
= 1.(a₁ – a₂) + 2(a₂ – a₃) + 3.(a₃ – a₄) + ………… + (k – 1)(aₖ₋₁ – aₖ) + k(aₖ – aₖ₊₁) …………. (i)

Here aₖ₊₁ = Number of students who gave wrong answer to (k+1) or more question
& as per question no student gave more than k wrong answer
∴ aₖ₊₁ = 0
Hence equation (i) can again be written as:-
⟹ a₁ + a₂ + a₃ + ………… + aₖ

∴ total number of wrong answers
= a₁ + a₂ + a₃ + ………… + aₖ Answer

28. In a certain test, 2⁽ᵏ⁻ᶦ⁾ students gave wrong answers to at least i questions, where i = 1, 2, 3, ………., k & no student gave more than k wrong answers. what is the total number of wrong answers given?
Solution:-

From previous question
total number of wrong answers = a₁ + a₂ + a₃ + …………. + aₖ

29. 30 players from India, Canada & Russia participatedd in world chess tournament. Number of players from india is 3 more than the number of players from canada. In tournament players from same country are not allowed to play against each other & no two player plays against each other more than once. Then find
(i). The maximum number of matches that was played in the tournament
(ii). If number of matches is maximum then what is the ratio of number of players from India & Russia ?
Solution:- 

Let number of players from Canada = x
∴ Number of players from India = x + 3
∴ Number of players from Russia = 30 – (x + x + 3)
= 27 – 2x
now from the given condition number of matches are as follows:-
⟹Number of matches between players of India & Canada is = ˣC₁ × ⁽ˣ⁺³⁾C₁ = x.(x+3)
⟹Number of matches between players of Canada & Russia is = ˣC₁ × ⁽²⁷⁻²ˣ⁾C₁ = x.(27 – 2x)
⟹ Number of matches between players of India & Russia is = ⁽ˣ⁺³⁾C₁ × ⁽²⁷⁻²ˣ⁾C₁ = (x+3).(27-2x)

So total number of matches
= x.(x+3) + x.(27-2x) + (x+3).(27-2x)
= x² + 3x + 27x – 2x² + 27x + 81 – 2x² – 6x
= -3x² + 51x + 81

So total number of matches is ( -3x² + 51x + 81) & it will attain a maximum value at
x =  \(\frac{{ – 51}}{{ – 6}}\)  = \(\frac{{17}}{2}\)  = 8.5
but x must be integers
So possible value of x is 8 or 9
⦿ When x = 8 then total number of matches
= -3(8)² + 51×8 + 81
= 297

⦿ When x = 9 then total number of matches
= -3(9)² + 51×9 + 81
= 297
Hence maximum number of matches = 297 when x=8 or 9 Answer

(ii).
⦿ when x=8 then players from India & Russia is 11, 11. Hence required ration = 11 ∶ 11 = 1 ∶ 1
⦿ when x=9 then players from India & Russia is 12 & 9. Hence required ratio = 12 ∶ 9 = 4 ∶ 3

30. A committee of 5 men & 5 women is to be selected from 9 men & 10 women.
(i). In how many ways this can be done
(ii). In how many ways this can be done if a particular woman is always selected
(iii). In how many ways this can be done if a particular woman is always selected and a man is always rejected?
Solution:-

(i). 
5 men can be selected out of 9 men in ⁹C₅ ways
⟹ 5 women can be selected out of 10 women in ¹⁰C₅ ways
Hence number of ways to select 5 men & 5 women is = ⁹C₅×¹⁰C₅      Answer

(ii).
if a particular woman is always selected then this can be done in ⁹C₄ ways
Hence number of ways =  ⁹C₅×⁹C₄    Answer

(iii).
If a particular woman is always selected then this can be done in ⁹C₄ ways
⟹ if a particular man is always rejected then this can be done in ⁸C₅ ways
Hence number of ways = ⁹C₄×⁸C₅   Answer

✅ Well done on completing Set 3!

Continue practising with Selection and Combination Questions 31 to 40 → Set 4 or revisit the Selection and Combination Concept Page to strengthen your formulas and tricks before moving ahead.

Consistent practice is the key to mastering Selection and Combination for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and international exams including GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests. Want to understand the concept better? Read about Combination (Mathematics) on Wikipedia before attempting the next set.

This page is part of our complete series of Selection and Combination Question with solutions for competitive exams — covering every question type from basic to advanced so you can build speed, accuracy and confidence. Practising these questions regularly will also strengthen your core LCM and HCF concept before your exam day.