Difference between revisions of "2007 AIME II Problems/Problem 1"
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| − | There are | + | There are 7 different characters that can be picked, with 0 being the only number that can be repeated twice. |
*If <math>0</math> appears 0 or 1 times amongst the sequence, there are <math>\frac{7!}{(7-5)!} = 2520</math> sequences possible. | *If <math>0</math> appears 0 or 1 times amongst the sequence, there are <math>\frac{7!}{(7-5)!} = 2520</math> sequences possible. | ||
Revision as of 20:16, 21 January 2020
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 N/
.
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 
.
appears 0 or 1 times amongst the sequence, there are 
sequences possible.
places to place the 
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
