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GATE Probability & Combinatorics Questions
Practise probability & combinatorics (mathematics) for the Graduate Aptitude Test in Engineering: 10 multiple-choice questions, each with the correct answer and a worked explanation. Click an option to check it, or “New questions” for another set.
Probability & Combinatorics practice
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Q1.A card is drawn from a well-shuffled deck of 52. What is the probability that it is a red card?
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Answer: C — 1/2
There are 26 such cards in 52, so P = 26/52 = 1/2.
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Q2.4 fair coins are tossed. What is the probability of at least one head?
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Answer: D — 15/16
P(no head) = (1/2)^4 = 1/16; P(at least one head) = 1 − 1/16 = 15/16.
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Q3.In how many ways can a committee of 3 be chosen from 12 people?
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Answer: B — 220
Order does not matter → combinations: 12C3 = 12! / (3!·9!) = 220.
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Q4.In how many ways can a committee of 2 be chosen from 7 people?
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Answer: A — 21
Order does not matter → combinations: 7C2 = 7! / (2!·5!) = 21.
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Q5.In how many ways can a committee of 3 be chosen from 6 people?
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Answer: C — 20
Order does not matter → combinations: 6C3 = 6! / (3!·3!) = 20.
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Q6.Two fair dice are thrown. What is the probability that the sum is 4?
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Answer: A — 1/12
3 of the 36 equally likely outcomes give a sum of 4, so P = 3/36 = 1/12.
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Q7.In how many ways can the letters of the word ORANGE be arranged?
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Answer: A — 720
All 6 letters are different, so arrangements = 6! = 720.
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Q8.3 fair coins are tossed. What is the probability of at least one head?
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Answer: A — 7/8
P(no head) = (1/2)^3 = 1/8; P(at least one head) = 1 − 1/8 = 7/8.
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Q9.In how many ways can the letters of the word SQUARE be arranged?
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Answer: A — 720
All 6 letters are different, so arrangements = 6! = 720.
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Q10.In how many ways can a committee of 4 be chosen from 10 people?
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Answer: A — 210
Order does not matter → combinations: 10C4 = 10! / (4!·6!) = 210.
How to approach probability & combinatorics questions
Probability = favourable ÷ total outcomes. List small sample spaces (two dice = 36 outcomes). Use nPr when order matters and nCr when it does not; divide by repeats for repeated letters.
About GATE
Graduate Aptitude Test in Engineering — conducted by IISc Bengaluru and the IITs (organising institute rotates). Tests engineering and science graduates for M.Tech admissions and PSU recruitment; every paper includes General Aptitude.
Original practice questions written for this site in the style and difficulty of each exam’s syllabus. They are not past papers and the site is not affiliated with any exam body. Exam patterns, marking schemes and syllabi change — always check the latest official notification.
GATE practice topics
| Topic | Section | Question source |
|---|---|---|
| Algebra | Mathematics | Calculated (new numbers every set) |
| Probability & Combinatorics | Mathematics | Calculated (new numbers every set) |
| Physics | Science | Calculated + question bank |
| Logical Reasoning | Reasoning | Calculated (new numbers every set) |
| Vocabulary & Idioms | English | Question bank |