Question

A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in $2007.$ No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in $2007.$ A set of plates in which each possible sequence appears exactly once contains $N$ license plates. Find $\frac{N}{10}.$

Answer

Solution:

The available characters are the seven distinct symbols A, I, M, E, $2,$ $0,$ $7,$ where $0$ may be used up to twice and every other character at most once. Sequences using at most one $0$ consist of five distinct characters chosen from the seven, in order: $7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 = 2520.$

Sequences with two $0$ 's: choose the two positions for the $0$ 's in $\binom{5}{2} = 10$ ways, then fill the remaining three positions with distinct characters from the other six in $6 \cdot 5 \cdot 4 = 120$ ways, for $1200$ sequences.

Thus $N = 2520 + 1200 = 3720,$ and $\frac{N}{10} = 372.$

2007 AIME II Problem 1

Solution:

Problem 1 in Other Years