AMC 8 Daily Practice Round 9

Complete problem set with solutions and individual problem pages

Problem 24 Easy

Among five pairs of distinct shoes, four shoes are randomly selected. How many ways can at least one matching pair be formed?

  • A.

    130

  • B.

    178

  • C.

    194

  • D.

    205

  • E.

    210

Answer:A

The total number of ways to choose any 4 shoes from 10 shoes is:   C(10,4) = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210

The number of ways to select 4 shoes without forming any complete pair is:

- Choose 4 different pairs out of the 5 available:     C(5,4) = \frac{5!}{4!(5-4)!} = 5

- For each selected pair, choose one shoe (2 choices per pair):     2^4 = 16

- Total ways to pick 4 shoes without forming a pair:     5 \times 16 = 80

Thus, the number of ways to select 4 shoes with at least one matching pair is:   210 - 80 = 130

The answer is \text{A}.