Selection & Combination Questions with Solutions — Set 1 (Q1-Q10)

This set contains Selection and Combination Questions with Solutions — Set 1 (Q1-Q10) 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.

1. In a school 20 students gave gift to each other, then find the total number of gifts.

2. In how many ways at least 6 players can be selected from a group of 8 players.

3. In how many ways Ramesh can invite 6 friends out of his 15 friends for a dinner.

4. In how many ways at least 6 student be selected out of 12 students?

5. In how many ways 10 friends can be invited out of 20 friends for dinner if Ramesh & rahul are always invited.

6. A work can be compelted by 5 men or 7 women in 4 days. In how many ways labour force can be selected if work has to be finished in 4 days & available labour force is 12 men & 10 women? If labour force has only men or only women.

7. Out of 5 men & 10 women a team of 6 is to be formed such that at least one men is included in the team. In how many ways such a team can be formed?

8. Out of 4 boys & 6 girls a team of 5 person is to be formed such that the team should have at least 1 boy and two boys named Rahul & Ankit refuse to work together. In how many ways team can be formed?

9. A fruit basket has n identical mangoes, n identical oranges & n distinct fruit. what is the number of ways in which n fruits can be selected from the fruit basket?

10. Find the number of ways in which three numbers in A.P. can be selected from a set of n integers {1, 2, 3, ………..n}.

1. In a school 20 students gave gift to each other, then find the total number of gifts.
Solution:-

2 student can be selected out of 20 students in ²⁰C₂ ways.
& in each pair gifts given are 2.
Hence total number of gifts = 2.²⁰C₂ Answer

2. In how many ways at least 6 players can be selected from a group of 8 players.
Solution:-

( ⁸C₆ + ⁸C₇ + ⁸C₈ ) ways
= 28 + 8 + 1
= 37 ways Answer

3. In how many ways Ramesh can invite 6 friends out of his 15 friends for a dinner.
Solution:-

¹⁵C₆ ways
= 5005 ways Answer

4. In how many ways at least 6 student be selected out of 12 students?
Solution:-

¹²C₆ + ¹²C₇ + ¹²C₈+ ¹²C₉ + ¹²C₁₀ + ¹²C₁₁ + ¹²C₁₂     ways
= 924 + 792 + 495 + 220 + 66 + 12 + 1
=  2510 Answer

5. In how many ways 10 friends can be invited out of 20 friends for dinner if Ramesh & rahul are always invited.
Solution:-

¹⁸C₈ ways Answer

6. A work can be compelted by 5 men or 7 women in 4 days. In how many ways labour force can be selected if work has to be finished in 4 days & available labour force is 12 men & 10 women? If labour force has only men or only women.
Solution:-

case(i): when work is completed by men then number of ways of selecting labour force i.e. 5 men out of 12 men is 
            ¹²C₅ = 792
case(ii): when work is completed by women then number of ways of selecting labour force i.e. 7 women out of 10 women is 
            ¹⁰C₇ = 120

Hence required number of ways = 792 + 120 = 912 ways Answer

7. Out of 5 men & 10 women a team of 6 is to be formed such that at least one men is included in the team. In how many ways such a team can be formed?
Solution:-

5×252 + 10×210 + 10×120 + 5×45 + 1×10
= 4795 ways Answer

Method(2):

Required number of ways = 
number of ways of selecting 6 out of 15 people – number of ways of selecting only women (6) out of 10 women
= ¹⁵C₆ – ¹⁰C₆
= 5005 – 210
= 4795 ways Answer

8. Out of 4 boys & 6 girls a team of 5 person is to be formed such that the team should have at least 1 boy and two boys named Rahul & Ankit refuse to work together. In how many ways team can be formed?
Solution:-

= 4×15 + 5×20 + 2×15
= 190 ways Answer

9. A fruit basket has n identical mangoes, n identical oranges & n distinct fruit. what is the number of ways in which n fruits can be selected from the fruit basket?
Solution:-

Hence total number of ways of selection of n fruits from fruit basket = 
= 1×ⁿCₙ + 2×ⁿCₙ₋₁ + 3×ⁿCₙ₋₂ + ………. + (r+1)×ⁿCᵣ + ……… + (n+1)×ⁿC₀
= 1×ⁿC₀ + 2×ⁿC₁ + 3×ⁿC₂ + ………… + (r+1)×ⁿCₙ₋ᵣ + ……… + (n+1)ⁿCₙ

10. Find the number of ways in which three numbers in A.P. can be selected from a set of n integers {1, 2, 3, ………..n}.
Solution:-
From the given set let 3 numbers in A.P. be a, b and c.
then b – a = c – b

∴ a + c is even & this can happen by 2 ways
(i) Both of them are even
(ii) Both of them are odd
⦿  when n is even then set has \(\frac{n}{2}\) even & \(\frac{n}{2}\) odd numbers.

From this set two even numbers can be selected in \(^{\frac{n}{2}}{C_2}\)  ways.
Similarly two odd numbers can be selected in \(^{\frac{n}{2}}{C_2}\) ways.
∴ number of ways when n is even
= \(^{\frac{n}{2}}{C_2}{ + ^{\frac{n}{2}}}{C_2}\)

✅ Well done on completing Set 1!

Continue practising with Selection and Combination Questions 11 to 20 → Set 2 or revisit the Selection and Combination Concept Page to strengthen your formulas and tricks before moving ahead.

Consistent practice is the key to mastering LCM and HCF 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 LCM and HCF questions 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.