AMC 12 Combinatorics & Probability 2

Question 1

How many distinct arrangements are there of the letters in BANANA?

Accepted Answer

60

Answer

6!/(3!·2!) = 720/12 = 60.

Question 2

Two fair dice are rolled. What is the probability that the sum is 7?

Accepted Answer

1/6

Answer

Six of the 36 outcomes sum to 7, giving 6/36 = 1/6.

Question 3

How many ordered triples of nonnegative integers (x, y, z) satisfy x + y + z = 10?

Accepted Answer

66

Answer

By stars and bars, C(12,2) = 66.

Question 4

How many subsets of {1, 2, ..., 10}, including the empty set, contain no two consecutive integers?

Accepted Answer

144

Answer

The count for {1..n} is the Fibonacci number F(n+2), and F(12) = 144.

Question 5

How many permutations of 5 distinct letters leave no letter in its original position?

Accepted Answer

44

Answer

The derangement number D5 = 5!(1 − 1 + 1/2 − 1/6 + 1/24 − 1/120) = 44.

AMC 12 - - American Mathematics Contest 12 Practice Test