This set contains Selection and Combination Questions with Solutions — Set 6 (Q51-Q60) 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.
Selection & Combination Questions 51 to 60 with Solutions
51. In an examination the question paper has three sections namely physics, chemistry & Biology containing 5, 6 & 7 questions respectively. As per the rule a student has to answer minimum 3 questions from each section. In how many ways a student can attempt 10 questions?
52. In an exam a question paper has 10 questions. What is the total number of ways that a student can answer this paper?
53. Find the number of factors of 1800.
54. How many factors of 1800 other than 1800 are 4 digit number?
55. A class has 10 students 7 are boys and 3 are girls. In how many ways students can be selected for a project.
56. In a class of 12 students 7 are boy & rest are girls. In how many ways students can be selected for the project such that at least 1 boy & 1 girl is selected.
57. In a class of 12 students 7 are boys and rest are girls. In how many ways students can be selected for a project such that at least 2 boys & 2 girls are selected?
58. There are 8 questions in an examination. A student has to answer at least four question to pass the exam, in how many ways he can fail in the exam?
59. A fruit basket has 5 bananas, 6 Lemon & 5 different type of fruits. In how many ways one can select fruits from this fruit basket?
60. In a fruit basket 5 mangoes, 6 banana & 4 Lemons are kept. In how many ways one can select fruits from this fruit basket such that at least one fruit of each type is always selected?
Selection & Combination Questions 51 to 60 — Step-by-Step Solutions
51. In an examination the question paper has three sections namely physics, chemistry & Biology containing 5, 6 & 7 questions respectively. As per the rule a student has to answer minimum 3 questions from each section. In how many ways a student can attempt 10 questions?
Solution:-
As per question we have following 3 cases:-
Hence total number of ways is 7000 + 5250 + 3500 = 15750 ways Answer
52. In an exam a question paper has 10 questions. What is the total number of ways that a student can answer this paper?
Solution:-
Each question has two ways to answer it either answered or unanswered
∴ total number of ways to attempt question paper = 2¹⁰ = 1024 ways
this also include the 1 case when none of the questions is answered i.e. all questions are answered.
Hence total number of ways = 1024 – 1
= 1023 ways Answer
53. Find the number of factors of 1800.
Solution:-
1800 = 18×100 = 2×9×4×25 = 2³×3²×5²
Hence number of factors = (3+1).(2+1).(2+1)
= 4×3×3
= 36 Answer
54. How many factors of 1800 other than 1800 are 4 digit number?
Solution:-
1800 = 2³×3²×5²
So except 1800 itself other largest factor is = \(\frac{{1800}}{2}\) = 900 → three digits
Hence number of four digit factors = 0 Answer
55. A class has 10 students 7 are boys and 3 are girls. In how many ways students can be selected for a project.
Solution:-
Every student (either boy or girl) has 2 ways of selection → either selected or rejected.
∴ students can be selected in 2⁷×2³ = 2¹⁰ = 1024 ways
& this also includes the case when none of the students is selected.
Hence total number of ways = 1024 – 1
= 1023 Answer
56. In a class of 12 students 7 are boy & rest are girls. In how many ways students can be selected for the project such that at least 1 boy & 1 girl is selected.
Solution:-
Nuber of ways of selecting one or more boys is (2⁷ – 1) = 127 ways
Number of ways of selecting one or more girl is (2⁵ – 1) = 31 ways
Hence required number of ways is 127×31 = 3937 Answer
57. In a class of 12 students 7 are boys and rest are girls. In how many ways students can be selected for a project such that at least 2 boys & 2 girls are selected?
Solution:-
Number of ways of selecting at least 2 boys
= ⁷C₂ + ⁷C₃ + ⁷C₄ + ⁷C₅ + ⁷C₆ + ⁷C₇
= (⁷C₀ + ⁷C₁ + ⁷C₂ + ⁷C₃ + ⁷C₄ + ⁷C₅ + ⁷C₆ + ⁷C₇) – ⁷C₀ – ⁷C₁
= 2⁷ – 1 – 7
= 120
Similarly, number of ways of selecting at least 2 girls = ⁵C₂ + ⁵C₃ + ⁵C₄ + ⁵C₅
= (⁵C₀ + ⁵C₁ + ⁵C₂ + ⁵C₃ + ⁵C₄ + ⁵C₅) – (⁵C₀ + ⁵C₁)
= 2⁵ – 1 – 5
= 26
Hence total number of ways = 120×26
= 3120 Answer
58. There are 8 questions in an examination. A student has to answer at least four question to pass the exam, in how many ways he can fail in the exam?
Solution:-
Student can fail by answering 0 or 1 or 2 or 3 questions in the exam & this can be done in ⁸C₀ + ⁸C₁ + ⁸C₂ + ⁸C₃ ways.
= 1 + 8 + 28 + 56
= 93 ways Answer
59. A fruit basket has 5 bananas, 6 Lemon & 5 different type of fruits. In how many ways one can select fruits from this fruit basket?
Solution:-
we know that the total number of ways of selecting one or more things from ‘p’ identical things of one type, ‘q’ identical things of another type & ‘n’ distinct things is
= (p+1)(q+1)2ⁿ – 1
= (5+1)(6+1)2⁵ – 1
= 1343 ways Answer
60. In a fruit basket 5 mangoes, 6 banana & 4 Lemons are kept. In how many ways one can select fruits from this fruit basket such that at least one fruit of each type is always selected?
Solution:-
mangoes can be selected in 5 ways ( either 1 or 2 or 3 or 4 or 5 mangoes)
Similarly, bananas & Lemons can be selected in 6 & 4 ways.
So total number of ways = 5×6×4
= 120 ways Answer
✅ Well done on completing Set 6!
Continue practising with Selection and Combination Questions 61 to 70 → Set 7 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.
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