IMO Combinatorics and Counting 1 — Questions and Answers
Question 1: In how many ways can you arrange 3 books on a shelf?
- 3
- 6 (Correct answer)
- 9
- 12
Correct answer: 6
The number of arrangements = 3! = 3 × 2 × 1 = 6.
Question 2: A coin is tossed 3 times. How many possible outcomes are there?
- 4
- 6
- 8 (Correct answer)
- 10
Correct answer: 8
Each toss has 2 outcomes; total = 2³ = 8.
Question 3: How many 2-digit numbers can be formed using digits 1, 2, 3, 4 (without repetition)?
- 8
- 10
- 12 (Correct answer)
- 16
Correct answer: 12
There are 4 choices for the first digit and 3 for the second: 4 × 3 = 12.
Question 4: In how many ways can a committee of 2 be chosen from 5 people?
- 5
- 8
- 10 (Correct answer)
- 20
Correct answer: 10
C(5,2) = 5!/(2! × 3!) = (5 × 4)/(2 × 1) = 10.
Question 5: How many ways can first, second, and third prizes be awarded to 6 contestants?
- 18
- 60
- 90
- 120 (Correct answer)
Correct answer: 120
P(6,3) = 6!/(6−3)! = 6 × 5 × 4 = 120.
Question 6: A menu has 4 starters and 3 main courses. How many different meals (one starter + one main) are possible?
- 7
- 10
- 12 (Correct answer)
- 14
Correct answer: 12
Total combinations = 4 × 3 = 12 by the multiplication principle.
IMO Combinatorics and Counting 1
In how many ways can you arrange 3 books on a shelf?