IMO Practice Test — Permutations and Combinations
6 Questions • 20 min • Olympiad level
20:00
Question 1 of 6
hard
If nC2 = 45, then n is:
9
10
11
12
Explanation: n(n−1)/2 = 45 gives n = 10.
Question 2 of 6
hard
How many rectangles (including squares) can be formed on a standard 8 × 8 chessboard?
1296
204
64
256
Explanation: An 8 × 8 board has 9 vertical lines and 9 horizontal lines. A rectangle needs 2 vertical and 2 horizontal lines. 9C2 × 9C2 = 36 × 36 = 1296.
Question 3 of 6
hard
How many distinct arrangements are possible for the word MATHEMATICS?
4989600
9979200
2494800
1247400
Explanation: 11!/(2!2!2!) = 4989600.
Question 4 of 6
hard
How many 4-member committees can be formed from 6 men and 5 women if exactly 2 women are included?
150
200
120
180
Explanation: 5C2 × 6C2 = 10 × 15 = 150.
Question 5 of 6
hard
Find the number of ways to distribute 10 identical apples among 3 children such that each child gets at least one apple.
36
66
45
28
Explanation: This is equivalent to finding the number of positive integer solutions to x + y + z = 10. The formula is (n-1)C(r-1) = (10-1)C(3-1) = 9C2 = 36.
Question 6 of 6
hard
There are 12 points in a plane, out of which 5 are collinear. Find the number of triangles that can be formed using these points as vertices.
210
220
200
190
Explanation: Total possible triangles if none were collinear = 12C3. Triangles lost due to collinear points = 5C3. Total = 12C3 - 5C3 = 220 - 10 = 210.