2022 AMC 10B Problems/Problem 3

Problem

How many three-digit positive integers have an odd number of even digits?

$\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$

Solution

There are only $2$ ways for an odd number of even digits: $1$ even digit or all even digits.

Case 1: $1$ even digit

There are $5 \cdot 5 = 25$ ways to choose the odd digits, $5$ ways for the even digit, and $3$ ways to order the even digit. So, $25 \cdot 5 \cdot 3 = 375$ . However, there are $5 \cdot 5= 25$ ways that the hundred's digit is $0$ and we must subtract this from $375$ , leaving us with $350$ ways.

Case 2: all even digits

There are $5 \cdot 5 \cdot 5 = 125$ ways to choose the even digits, and $5 \cdot 5 = 25$ ways where the hundred's digit is $0$ . So, $125-25=100$ .

Adding up the cases, the answer is $100+350=\boxed{\textbf{(D) }450}$ .

~MrThinker

2022 AMC 10B (Problems • Answer Key • Resources)
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2022 AMC 10B Problems/Problem 3

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