Solution: 2019 Winter Final - 1

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 5 letters $c$?

(a)

${85 \choose 5} \cdot 3^{85}$

(b)

${85 \choose 5} \cdot 3^{80}$

(c)

${85 \choose 5} \cdot 4^{80}$

(d)

${85 \choose 5} \cdot 4^{85}$

We choose 5 out of the 85 spots for the c’s to be placed: $ \binom{85}{5}$

Each of the remaining 80 characters have 3 possible letters to be, which are $a$, $b$, or $d$: $ 3 \cdot 3 \cdot \ldots \cdot 3 = 3^{80}$

Multiplying everything up, we get $ \binom{85}{5} \cdot 3^{80} $