Solution: 2014 Fall Final - 1

Author: Michiel Smid

Question

A password consists of 13 characters, each character being one of the ten digits $0,1,2,\dots,9$.
A password must contain exactly one odd digit. How many passwords are there?

(a)

$13 \cdot 5^{13}$

(b)

$13 \cdot 9^{12}$

(c)

$13 \cdot 5^{12}$

(d)

$13 \cdot 5 \cdot 9^{12}$

Solution

First, we choose the odd digit from $ { 1,3,5,7,9 } $: $5 \text{ ways} $

Then, we choose which position to place the odd digit: $ \binom{13}{1} \text{ ways} $

Finally, we choose which of the following 5 event digits for the remaining 12 digits: $ 5^{12} \text{ ways} $

$ 5 \cdot \binom{13}{1} \cdot 5^{12} $

$ 13 \cdot 5^{13} $