2007 AIME II Problem 1

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

Answer: 372

Concepts:permutations casework multiplication principle

Difficulty rating: 1890

Small hint:

Split into cases by how many

0

’s the sequence uses; every other character can appear at most once

Big hint:

With two

0

’s, place them in

\binom{5}{2}

ways and fill three spots from the six other characters

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.