If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is :

Solution

Maths sir · 4 steps

Pick the two addresses first, then share the three letters between them so neither address is left empty.

1. Total outcomes. Each letter can go to any of the 5 addresses, independently. So there are 5³ = 125 ways to post the three letters.
2. Choose two addresses. Exactly two addresses are used, so first choose which two: C(5, 2) = 10 ways.
3. Use both addresses. Each letter goes to one of these 2 addresses, giving 2³ = 8 ways. Remove the 2 ways where all three letters go to the same address, which leaves 6 ways where both addresses get at least one letter.

   P = (C(5, 2) × (2³ - 2))/5³ = (10 × 6)/125 = 60/125 = 12/25

4. So. The probability that the letters reach exactly two addresses is 12/25, which is option (b).

The idea to keep. For exactly k boxes used, choose the k boxes, then count only the onto fillings of those boxes, so that none of them is left empty. Divide by the total, where every object picks freely.

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