Solution: 2018 Fall Final - 1

Author: Michiel Smid

Question

Consider strings of length 70, in which each character is one of the letters $a,b,c$. How many such strings have exactly 1 letter $c$?

(a)

$70 \cdot 2^{69}$

(b)

$70 \cdot 2^{70}$

(c)

$70 \cdot 3^{70}$

(d)

$70 \cdot 3^{69}$

First, we choose the position of the letter $c$: $ \binom{70}{1} = 70$

Then, we choose the positions of the letters $a$ and $b$ for th remaining 69 positions: $ 2^{69} $

$ |C| = 70 \cdot 2^{69} $