Solution: 2019 Fall Final - 2

Author: Michiel Smid

Question

Consider strings of length 85, in which each character is one of the letters $a,b,c,d$.
How many such strings have exactly 15 letters $a$ and exactly 30 letters $d$?

(a)

${85 \choose 15} \cdot {70 \choose 30} \cdot 3^{40}$

(b)

${85 \choose 15} \cdot {70 \choose 30} \cdot 2^{40}$

(c)

${85 \choose 15} \cdot {70 \choose 30} \cdot 4^{40}$

(d)

None of the above.

Solution

We choose 15 of the 85 positions to be a: $ \binom{85}{15} $

We choose 30 of the 70 remaining positions to be d: $ \binom{70}{30} $

The remaining 40 positions can be any of the 2 letters: $ 2^{40} $

Thus, the total number of strings is $ \binom{85}{15} \cdot \binom{70}{30} \cdot 2^{40} $