Mock AIME 1 Pre 2005 Problems/Problem 7
Problem
Let denote the number of permutations of the -character string such that
- None of the first four letters is an .
- None of the next five letters is a .
- None of the last six letters is a .
Find the remainder when is divided by .
Solution
Let there be As amongst the five numbers in the middle (those mentioned in condition [2]). There are As amongst the last six numbers then. Also, there are Cs amongst the middle five numbers, and so there are Cs amongst the first four numbers.
Thus, there are ways to arrange the first four numbers, ways to arrange the middle five numbers, and ways to arrange the last six numbers. Notice that leads to a contradiction, so the desired sum is And .
See also
Mock AIME 1 Pre 2005 (Problems, Source) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |