Solution: 2019 Winter 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)

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

(c)

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

(d)

None of the above.

Solution

We choose 15 of the 85 spots for the $a$‘s: $\binom{85}{15}$

Then we choose 30 of the remaining 70 spots for the $d$‘s: $\binom{70}{30}$

The remaining 40 spots can be $c$ or $d$: $ 2 \cdot 2 \cdot \ldots \cdot 2 = 2^{40} $

Multiplying everything together, we get $\binom{85}{15} \cdot \binom{70}{30} \cdot 2^{40}$