Question

There are 5 pairs of shoes in a cupboard from which 4 shoes are picked at random. The probability that there is at least one pair is:

• A. $\frac821$
• B. $\frac1121$
• C. $\frac1321$
• D. $\frac1231$

Hint:

When finding “at least one” probabilities, it’s often easier to subtract the “none” case from 1.

Answer 1

The Correct Option is C

Step 1: Total number of shoes = $5$ pairs × $2$ shoes each = $10$ shoes.
We are selecting 4 shoes at random.
Step 2: Total ways to choose any 4 shoes from 10: \[ \binom104 = 210 \] Step 3: We need probability of at least one pair.
It’s easier to find the complement — probability of no pairs — and subtract from 1.
Step 4: For no pairs: Choose 4 different pairs from the 5 available. Ways to choose the pairs: \[ \binom54 = 5 \] From each chosen pair, select 1 shoe (left or right): \[ 2^4 = 16 \] Thus, total ways with no pairs: \[ 5 \times 16 = 80 \] Step 5: Probability of no pairs: \[ \frac80210 = \frac821 \] Step 6: Probability of at least one pair: \[ 1 - \frac821 = \frac1321 \] Step 7: Thus, the probability is $\mathbf\frac1321$.