There are 10 pairs of shoes in a cupboard from which 4 shoes are taken at random. Then the probability of getting

Given 10 pairs of shoes (20 individual shoes) in a cupboard, we randomly select 4 shoes. Let's calculate the probabilities for each scenario:

Total Ways to Select 4 Shoes

The total number of ways to choose 4 shoes from 20 is calculated using combinations:

Total Ways:

20C₄ = 4845

a) Probability of Selecting No Pair

To select 4 shoes with no pairs, each shoe must come from a different pair.

• Selecting 4 Pairs from 10: 10C₄ = 210
• Choosing 1 Shoe from Each Selected Pair: 2⁴ = 16
• Total Ways: 210 × 16 = 3360

Probability:

P(No Pair) = 3360 / 4845 = 224 / 323

b) Probability of Selecting At Least One Pair

This is the complement of selecting no pair.

Probability:

P(At Least One Pair) = 1 - P(No Pair) = 1 - 224 / 323 = 99 / 323

c) Probability of Selecting Exactly Two Pairs

• Selecting 2 Pairs from 10: 10C₂ = 45
• Total Ways to Select 4 Shoes: 45 × 1 = 45

Probability:

P(Exactly Two Pairs) = 45 / 4845 = 3 / 323

d) Probability of Selecting Exactly One Pair

This can be found by subtracting the probabilities of other scenarios from 1.

Probability:

P(Exactly One Pair) = 1 - P(No Pair) - P(Exactly Two Pairs) = 1 - 224 / 323 - 3 / 323 = 96 / 323