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ACT Probability & Combinatorics Questions
Practise probability & combinatorics (mathematics) for the ACT: 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.In how many ways can the letters of the word GARDEN be arranged?
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Answer: A — 720
All 6 letters are different, so arrangements = 6! = 720.
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Q2.In how many ways can the letters of the word PENCIL be arranged?
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Answer: A — 720
All 6 letters are different, so arrangements = 6! = 720.
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Q3.4 fair coins are tossed. What is the probability of at least one head?
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Answer: A — 15/16
P(no head) = (1/2)^4 = 1/16; P(at least one head) = 1 − 1/16 = 15/16.
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Q4.In how many ways can a committee of 4 be chosen from 12 people?
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Answer: C — 495
Order does not matter → combinations: 12C4 = 12! / (4!·8!) = 495.
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Q5.2 fair coins are tossed. What is the probability of at least one head?
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Answer: B — 3/4
P(no head) = (1/2)^2 = 1/4; P(at least one head) = 1 − 1/4 = 3/4.
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Q6.In how many ways can the letters of the word MOTHER be arranged?
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Answer: B — 720
All 6 letters are different, so arrangements = 6! = 720.
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Q7.Two fair dice are thrown. What is the probability that the sum is 6?
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Answer: B — 5/36
5 of the 36 equally likely outcomes give a sum of 6, so P = 5/36 = 5/36.
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Q8.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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Q9.A card is drawn from a well-shuffled deck of 52. What is the probability that it is a heart?
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Answer: C — 1/4
There are 13 such cards in 52, so P = 13/52 = 1/4.
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Q10.A card is drawn from a well-shuffled deck of 52. What is the probability that it is an ace or a king?
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Answer: A — 2/13
There are 8 such cards in 52, so P = 8/52 = 2/13.
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 ACT
ACT — conducted by ACT, Inc.. US college admission test with English, Math and Reading, plus an optional Science section.
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.
ACT practice topics
| Topic | Section | Question source |
|---|---|---|
| Algebra | Mathematics | Calculated (new numbers every set) |
| Geometry & Mensuration | Mathematics | Calculated (new numbers every set) |
| Probability & Combinatorics | Mathematics | Calculated (new numbers every set) |
| English Grammar | English | Question bank |
| Physics | Science | Calculated + question bank |
| Chemistry | Science | Calculated + question bank |