IMO Combinatorics and Counting 2 — Questions and Answers
Question 1: What is C(6, 2)?
- 12
- 15 (Correct answer)
- 18
- 30
Correct answer: 15
C(6,2) = 6!/(2! × 4!) = (6 × 5)/(2 × 1) = 15.
Question 2: A die is rolled once. What is the probability of getting an even number?
- 1/6
- 1/3
- 1/2 (Correct answer)
- 2/3
Correct answer: 1/2
Even numbers on a die: 2, 4, 6 — that's 3 out of 6; probability = 3/6 = 1/2.
Question 3: How many 4-letter codes can be made from 26 letters if repetition is allowed?
- 26⁴ (Correct answer)
- 26 × 4
- 26!
- C(26,4)
Correct answer: 26⁴
With repetition allowed, each of the 4 positions has 26 choices: total = 26⁴ = 456,976.
Question 4: In how many ways can 4 people be seated in a row of 4 chairs?
- 12
- 16
- 20
- 24 (Correct answer)
Correct answer: 24
This is 4! = 4 × 3 × 2 × 1 = 24.
Question 5: A bag has 5 red and 3 blue balls. What is the probability of picking a blue ball?
- 3/5
- 3/8 (Correct answer)
- 5/8
- 1/3
Correct answer: 3/8
Total balls = 8; P(blue) = 3/8.
Question 6: How many ways can a team of 3 be selected from 7 players?
- 21
- 35 (Correct answer)
- 42
- 70
Correct answer: 35
C(7,3) = 7!/(3! × 4!) = (7 × 6 × 5)/(3 × 2 × 1) = 35.
IMO Combinatorics and Counting 2
What is C(6, 2)?