Mock AIME 1 Pre 2005 Problems/Problem 7


Let $N$ denote the number of permutations of the $15$-character string $AAAABBBBBCCCCCC$ such that

  1. None of the first four letters is an $A$.
  2. None of the next five letters is a $B$.
  3. None of the last six letters is a $C$.

Find the remainder when $N$ is divided by $1000$.


Let there be $k$ As amongst the five numbers in the middle (those mentioned in condition [2]). There are $4-k$ As amongst the last six numbers then. Also, there are $5-k$ Cs amongst the middle five numbers, and so there are $6-(5-k) = k+1$ Cs amongst the first four numbers.

Thus, there are ${4 \choose k+1}$ ways to arrange the first four numbers, ${5 \choose k}$ ways to arrange the middle five numbers, and ${6 \choose 4-k} = {6\choose k+2}$ ways to arrange the last six numbers. Notice that $k=4$ leads to a contradiction, so the desired sum is \[\sum_{k=0}^{3} {4\choose k+1}{5\choose k}{6\choose k+2} = 60 + 600 + 600 + 60 = 1320\] And $N \equiv \boxed{320} \pmod{1000}$.

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