2007 AIME II Problems/Problem 1

Problem

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 $\frac{N}{1000}$ .

Solution

There are 7 different characters that can be picked, with 0 being the only number that can be repeated twice.

If $0$ appears 0 or 1 times amongst the sequence, there are $\frac{7!}{(7-5)!} = 2520$ sequences possible.
If $0$ appears twice in the sequence, there are ${5\choose2} = 10$ places to place the $0$ s. There are $\frac{6!}{(6-3)!} = 120$ ways to place the remaining three characters. Totally, that gives us $10 \cdot 120 = 1200$ .

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

2007 AIME II (Problems • Answer Key • Resources)
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2007 AIME II Problems/Problem 1

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