Selection & Combination Questions with Solutions — Set 5 (Q41-Q50)

This set contains Selection and Combination Questions with Solutions — Set 5 (Q41-Q50) 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.

41. How many different sums can be formed with the following coins:⟶ one 1 rupee, one 50 paise, one 25 paise, one 10 paise & one 5 paise coins?

42. Given 6 different green dyes, 5 different blue dyes & 4 different red dyes, the number of combination of dyes which can be chosen taking at least one green & one blue dye is?

43. In how many ways 11 players can be selected out of 25 players if Ram & Shyam are always selected and Mohit & Ankit are always rejected?

44. In how many ways 8 distinct alphabets can be selected so that no vowel is selected?

45. In class (V) there are three sections A, B & C with number of students 50, 70 and 90 respectively while in class (vi) there are three sections A, B & C with number of students 70, 90 & 50 respectively. In how many ways 15 students can be selected either from class V or class VI equal from section A, B & C.

46. In an examination a student has to write 8 papers & has to pass in each paper to pass the exam. In how many ways a student can fail in the exam?

47. In the annual function of a school 10 boys & 20 girls danced such that in a particular dance only 1 boy & 1 girl participated. Total how many dance was performed?

48. In a dance competition, there are 10 participants. In each performance exactly two dancers will perform. Mr. Abhishek, a fan of this show has decided to see exactly 10 dance iperformance. In how many ways he can select the performance?

49. Consider a set x = {1, 2, 3, …….., 9, 10}. What is the number of pair {A, B} such that A ⊆ x and B ⊆ x also A ≠ B and A ∩ B = {2, 3, 5, 7}.

50. A committee of 12 students has to be selected from 18 students 6 from each management, sales & finance such that at least 2 student must be selected from each management, sales & finance and number of students from management is more than the number of students from sales which in turn is more than the number of students from finance. In how many ways this selection can be achieved?

41. How many different sums can be formed with the following coins:⟶ one 1 rupee, one 50 paise, one 25 paise, one 10 paise & one 5 paise coins?
Solution:-

Method (1).
Every coin has 2 ways of selection:⟶ either selected or not selected

= 2⁵ – 1
= 31 Answer

Method(2):

Number of different sums = ⁵C₁ + ⁵C₂ + ⁵C₃ + ⁵C₄ + ⁵C₅
                                               = 5 + 10 + 10 + 5 + 1
                                               = 31 Answer

42. Given 6 different green dyes, 5 different blue dyes & 4 different red dyes, the number of combination of dyes which can be chosen taking at least one green & one blue dye is?
Solution:-

one or more (at least one) green dye can be chosen in ⁶C₁ + ⁶C₂ + ⁶C₃ + ⁶C₄ + ⁶C₅ + ⁶C₆ = (2⁶ – 1) ways
One or more (at least one) blue dye can be chosen in ⁵C₁ + ⁵C₂ + ⁵C₃ + ⁵C₄ + ⁵C₅ = (2⁵ – 1) ways
Zero or more (Any number of) red dye can be chosen in ⁴C₀ + ⁴C₁ + ⁴C₂ + ⁴C₃ + ⁴C₄ = 2⁴ ways
So, total number of required selection
= (2⁶ – 1)×(2⁵ – 1)×2⁴
= 63×31×16
= 31248 ways Answer

43. In how many ways 11 players can be selected out of 25 players if Ram & Shyam are always selected and Mohit & Ankit are always rejected ?
Solution:-

44. In how many ways 8 distinct alphabets can be selected so that no vowel is selected?
Solution:-

total alphabets = 26
vowels = 5
consonants = 26 – 5 = 21

so we have to select 8 consonants out of 21 consonants & this can be done in ²¹C₈ ways       Answer

45. In class (V) there are three sections A, B & C with number of students 50, 70 and 90 respectively while in class (vi) there are three sections A, B & C with number of students 70, 90 & 50 respectively. In how many ways 15 students can be selected either from class V or class VI equal from section A, B & C.
Solution:-

Equal number of students means 5 students from each section.
case(i).
If student are selected from class V then number of ways is ⁵⁰C₅×⁷⁰C₅×⁹⁰C₅
case(ii).
If students are selected from class VI then the number of ways is ⁷⁰C₅×⁹⁰C₅×⁵⁰C₅

∴ Total number of ways = either case(i) OR case(ii)
= ⁵⁰C₅×⁷⁰C₅×⁹⁰C₅ + ⁷⁰C₅×⁹⁰C₅×⁵⁰C₅ Answer

46. In an examination a student has to write 8 papers & has to pass in each paper to pass the exam. In how many ways a student can fail in the exam?
Solution:-

For every paper there are 2 ways of selection → Pass or Fail.
So total number of ways is 2⁸ = 256 & out of these ways one way is also counted when student pass in every paper i.e. student pass the exam in 1 way.
So total number of ways that the student can fail is = 256 – 1
= 255 Answer

47. In the annual function of a school 10 boys & 20 girls danced such that in a particular dance only 1 boy & 1 girl participated. Total how many dance was performed?
Solution:-

In a dance we have to select one boy out of 10 boys in ¹⁰C₁ = 10 ways, and one girl oout of 20 girls in ²⁰C₁ = 20 ways,
Hence total number of dance performed = 10×20
= 200 Answer

48. In a dance competition, there are 10 participants. In each performance exactly two dancers will perform. Mr. Abhishek, a fan of this show has decided to see exactly 10 dance iperformance. In how many ways he can select the performance?
Solution:-

Total number of dance performance is ¹⁰C₂ = 45
So Mr. Abhishek has to select 10 performances from these 45 performances
& that can be done in ⁴⁵C₁₀ ways      Answer

49. Consider a set x = {1, 2, 3, …….., 9, 10}. What is the number of pair {A, B} such that A ⊆ x and B ⊆ x also A ≠ B and A ∩ B = {2, 3, 5, 7}.
Solution:-

Since it is given that A ∩ B = {2, 3, 5, 7}, so remaining numbers {1, 4, 6, 8, 9, 10} have three option to go in either set A or set B or neither in set A nor in set B. So each member of set {1, 4, 6, 8, 9, 10} has 3 option of selection. So total number of ways is 3⁶ but since A ≠ B, so subtract the case when A = Φ & B = Φ i.e. A = B.
Hence required number of ways = 3⁶ – 1     Answer

50. A committee of 12 students has to be selected from 18 students 6 from each management, sales & finance such that at least 2 student must be selected from each management, sales & finance and number of students from management is more than the number of students from sales which in turn is more than the number of students from finance. In how many ways this selection can be achieved?
Solution:-

Hence total number of ways is 1800 + 225 = 2025 ways      Answer

✅ Well done on completing Set 5!

Continue practising with Selection and Combination Questions 51 to 60 → Set 6 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.