IMO Combinatorics and Counting

Question 1

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

Accepted Answer

6

Answer

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

Question 2

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

Accepted Answer

8

Answer

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)?

Accepted Answer

12

Answer

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?

Accepted Answer

10

Answer

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?

Accepted Answer

120

Answer

P(6,3) = 6!/(6−3)! = 6 × 5 × 4 = 120.

IMO - International Mathematics Olympiad Practice Test