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2022 AMC 10B Problem 3

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2022 AMC 10B Problem 3AMC 10 Step by Step
Name:                             Date: September 13, 2026
Basic Counting · difficulty 2 of 5 · about 2 min · position P1-10

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

  1. A)

    150150

  2. B)

    250250

  3. C)

    350350

  4. D)

    450450

  5. E)

    550550

Answer

D

Key insight

Fix the first two digits; among the ten possible units digits exactly five flip the parity of the even-digit count, so exactly half of all 900 numbers work.

Solution

Choose the hundreds and tens digits first: 9⋅10=909 \cdot 10 = 90 ways. Those two digits contain some number of even digits, either even or odd.

Now the units digit runs over 0,1,…,90, 1, \dots, 9: five even values and five odd values. If the first two digits already contain an odd number of even digits, the five odd units digits keep the count odd; if they contain an even number, the five even units digits make it odd. Either way exactly 55 of the 1010 units digits work.

Total: 90⋅5=45090 \cdot 5 = 450.

The answer is (D) 450\boxed{\textbf{(D)}\ 450}.

Why this works

The last digit acts as a "parity switch": whatever the earlier digits did, half of the choices for the final digit produce each parity. This pairing argument shows that exactly half of the 900900 three-digit numbers qualify, with no casework at all. Look for a free position that can flip the property you care about.

Alternative approach

Direct casework. Exactly one even digit: even hundreds digit with two odd digits, 4⋅5⋅5=1004 \cdot 5 \cdot 5 = 100; odd hundreds digit with exactly one of the other two even, 5⋅2⋅5⋅5=2505 \cdot 2 \cdot 5 \cdot 5 = 250. Exactly three even digits: 4⋅5⋅5=1004 \cdot 5 \cdot 5 = 100. Sum 100+250+100=450100 + 250 + 100 = 450.

The trap

Treating the hundreds digit like the others (it cannot be 0, so only 4 of its 9 values are even) when doing explicit casework.

Common mistakes

  • Treating the hundreds digit like the others (it cannot be 0, so only 4 of its 9 values are even) when doing explicit casework.
  • Forgetting the "three even digits" case and answering 350.

Techniques

Exploit symmetry to reduce work or pair up objects