2021 JMC 10 Problems/Problem 8

Problem

A positive integer is pretentious if it has both even and odd digits. For example, $74$ and $83$ are pretentious. How many pretentious three-digit numbers are odd?

$\textbf{(A) } 325 \qquad\textbf{(B) } 375 \qquad\textbf{(C) } 370 \qquad\textbf{(D) } 450 \qquad\textbf{(E) } 775$

Solution

Suppose an odd pretentious three-digit number is of the form $\underline{a}~\underline{b}~\underline{c},$ where $c$ equals $1,3,5,7,$ or $9.$ Both $a$ and $b$ can be even, be even and odd, odd and even, but cannot both be odd. Using complementary counting, there are $9 \cdot 10 = 90$ total choices for $(a,b)$ and $5 \cdot 5 = 25$ undesired cases (when both digits are odd), leaving $90 - 25 = 65$ desired pairs $(a,b)$. Thus, the answer is $65 \cdot 5 = 325$ such numbers.