Solution: 2019 Fall 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)

(\binom{85}{5}) \cdot 3^{80}

(b)

(\binom{85}{5}) \cdot 4^{85}

(c)

(\binom{85}{5}) \cdot 4^{80}

(d)

(\binom{85}{5}) \cdot 3^{85}

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 $(\binom{85}{5}) \cdot 3^{80}$