Difference between revisions of "2007 AIME II Problems/Problem 1"
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== Problem == | == 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 <math>2007</math>. No character may appear in a [[sequence]] more times than it appears among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains <math>N</math> license plates. Find <math>\frac{N}{ | + | 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 <math>2007</math>. No character may appear in a [[sequence]] more times than it appears among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains <math>N</math> license plates. Find <math>\frac{N}{1000}</math>. |
== Solution == | == Solution == |
Revision as of 18:19, 13 December 2010
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
2007 AIME II (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |