Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is :

Solution

Maths sir · 4 steps

Go through the complement: 0, 1 or 2 correct shirts, with the rest of the players fully mixed up.

1. Use the complement. Counting 3 or more correct directly is long. Instead, P(≥ 3) = 1 - P(0) - P(1) - P(2), where P(k) is the chance that exactly k players get their own shirt.
2. Exactly k correct. Choose the k lucky players in C(15, k) ways. The other 15-k must all be wrong, which is a derangement D₁₅₋ₖ. Divide by the 15! total arrangements, and use Dₘ/m! = 1 - 1/1! + 1/2! - … ≈ 1/e, which is extremely accurate for m ≥ 13.

   P(k) = (C(15, k) D₁₅₋ₖ)/15! = 1/k!·D₁₅₋ₖ/(15-k)! ≈ 1/(k! e)

3. Add the three. P(0) + P(1) + P(2) ≈ 1/e(1 + 1 + 1/2) = 5/2e ≈ 2.5 × 0.3679 = 0.9197.
4. So. P(≥ 3) ≈ 1 - 0.9197 = 0.0803, which is about 0.08, option (d).

The idea to keep. When n items are matched at random and n is large, the chance of exactly k correct is about 1/(e k!). For an 'at least' question, subtract the small cases from 1.

Answer from the NTA final answer key. Solution by PrepWiser.