Solution: 2019 Fall Final - 1
Author: Michiel SmidQuestion
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 $c$?
How many such strings have exactly 5 letters $c$?
(a)
${85 \choose 5} \cdot 4^{85}$
(b)
${85 \choose 5} \cdot 3^{85}$
(c)
${85 \choose 5} \cdot 3^{80}$
(d)
${85 \choose 5} \cdot 4^{80}$
Solution
We choose 5 of the 85 positions to be c: $ \binom{85}{5} $
The remaining 80 positions can be any of the 3 letters: $ 3^{80} $
Thus, the total number of strings is $ \binom{85}{5} \cdot 3^{80} $