COMC COMC Geometry and Combinatorics 2

Question 1

In how many ways can 5 distinct books be arranged on a shelf?

Accepted Answer

120

Answer

The number of permutations of 5 distinct objects is 5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 2

How many ways can a committee of 3 people be chosen from a group of 12?

Accepted Answer

220

Answer

C(12,3) = 12!/(3!×9!) = (12×11×10)/(3×2×1) = 220.

Question 3

How many 4-digit numbers can be formed using digits 1–9 without repetition?

Accepted Answer

3024

Answer

This is P(9,4) = 9 × 8 × 7 × 6 = 3024 ordered arrangements.

Question 4

If two fair six-sided dice are rolled, in how many ways can the sum equal 7?

Accepted Answer

6

Answer

The pairs are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1), giving exactly 6 ordered outcomes.

Question 5

In how many ways can 8 people be split into two unlabeled groups of 4?

Accepted Answer

35

Answer

Choose 4 people in C(8,4) = 70 ways, then divide by 2 because swapping the two groups gives the same split: 70/2 = 35.

COMC - Canadian Open Mathematics Challenge Practice Test