jmc

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 N10.

Accepted Answer

If a sequence contains no more than one 0, there are 7 6 5 4 3 = 2520 sequences formed from the characters A, I, M, E, 2, 0, and 7. If a sequence contains two 0's, the 0's can be placed in 52 = 10 ways, the remaining characters can be chosen in 63 = 20 ways, and those remaining characters can be arranged in 3! = 6 ways, for a total of 10 20 6 = 1200 sequences. Thus N = 2520 + 1200 = 3720, and N10= 372.