AMC12 Combinatorics and Probability 3

Question 1

The number of ways to distribute 5 identical balls into 3 distinct boxes (allowing empty boxes) is:

Accepted Answer

21

Answer

By stars and bars, C(5+3−1, 3−1) = C(7, 2) = 21.

Question 2

How many distinct diagonals does a convex octagon have?

Accepted Answer

20

Answer

Diagonals = C(8, 2) − 8 = 28 − 8 = 20.

Question 3

A function f: {1, 2, 3} → {1, 2, 3} is chosen at random. What is the probability it is a bijection?

Accepted Answer

2/9

Answer

There are 3³ = 27 total functions and 3! = 6 bijections, so P = 6/27 = 2/9.

Question 4

If 5 fair dice are rolled, what is the probability all show different faces?

Accepted Answer

5/54

Answer

P = (6 × 5 × 4 × 3 × 2)/6⁵ = 720/7776 = 5/54.

Question 5

In a group of 30 people, the probability that at least two share a birthday is approximately:

Accepted Answer

Greater than 70%

Answer

The birthday problem gives approximately 70.6% for 30 people.

AMC12 - American Mathematics Competition 12 Practice Test