2026-04-12

Problem — 2026-04-12

Medium

Mediumprobability

You have $5$ pairs of shoes ( $10$ shoes total) in a closet. You randomly pick $4$ shoes. What is the probability that you get at least one matching pair?

Express as a fraction

Show solution

Use the complement. To get NO matching pair, choose

4

of the

5

shoe types (

\binom{5}{4} = 5

ways), then pick one shoe (left or right) from each chosen pair (

2^4 = 16

ways). So no-match outcomes:

5 \times 16 = 80

. Total ways to pick

4

from

10

\binom{10}{4} = 210

P(\text{no match}) = \frac{80}{210} = \frac{8}{21}

. So

P(\text{at least one match}) = 1 - \frac{8}{21} = \frac{13}{21}