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Combinatorics and Counting Flashcards

6 cards from real IMO practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

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  1. In how many ways can you arrange 3 books on a shelf?

    Answer: 6

    The number of arrangements = 3! = 3 × 2 × 1 = 6.

  2. A coin is tossed 3 times. How many possible outcomes are there?

    Answer: 8

    Each toss has 2 outcomes; total = 2³ = 8.

  3. How many 2-digit numbers can be formed using digits 1, 2, 3, 4 (without repetition)?

    Answer: 12

    There are 4 choices for the first digit and 3 for the second: 4 × 3 = 12.

  4. In how many ways can a committee of 2 be chosen from 5 people?

    Answer: 10

    C(5,2) = 5!/(2! × 3!) = (5 × 4)/(2 × 1) = 10.

  5. How many ways can first, second, and third prizes be awarded to 6 contestants?

    Answer: 120

    P(6,3) = 6!/(6−3)! = 6 × 5 × 4 = 120.

  6. A menu has 4 starters and 3 main courses. How many different meals (one starter + one main) are possible?

    Answer: 12

    Total combinations = 4 × 3 = 12 by the multiplication principle.

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Sources & references

Exam facts on this page (fees, format, dates, passing scores) are checked against these official sources; the date is our last check. Editorial policy

  1. imo-official.org — Test delivery, eligibility, passing score, test dates, exam format · verified Oct 2026
  2. imo2027.hu — Exam dates, registration deadline · verified Oct 2026