2007 AIME II Problems/Problem 1

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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 . No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in . A set of plates in which each possible sequence appears exactly once contains license plates. Find .

Solution

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

• If appears 0 or 1 times amongst the sequence, there are sequences possible.
• If appears twice in the sequence, there are places to place the s. There are ways to place the remaining three characters. Totally, that gives us .

Thus, , and .

See also