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

Question

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:

Accepted Answer

Given: Three letters can each be posted independently to any one of $5$ different addresses.

Find: The probability that exactly $2$ addresses are used.

The total number of ways to post the $3$ letters is

5^3 = 125

Now count the favorable cases where exactly $2$ addresses are used.

First, choose $2$ addresses out of $5$ :

\binom{5}{2} = 10

For a fixed pair of addresses, the $3$ letters must be distributed so that both addresses get at least one letter.

Each letter can go to either of the $2$ chosen addresses, so total assignments are

2^3 = 8

But exclude the $2$ cases where all $3$ letters go to only one address. Hence favorable assignments for a fixed pair are

2^3 - 2 = 6

Therefore, total favorable outcomes are

\binom{5}{2}(2^3 - 2) = 10 \times 6 = 60

Hence, the required probability is

\frac{60}{125} = \frac{12}{25}

Therefore, the correct option is A.

Given: Three letters and 5 addresses. Find: The probability that exactly 2 addresses are used. Choose the 2 addresses first, then count all onto assignments from 3 letters to these 2 addresses: 52(2^3 - 2) The term 2^3 counts all assignments to the chosen pair, and subtracting 2 removes the two cases in which all letters go to a single address. So, Favorable = 52(2^3 - 2) = 10 6 = 60 And total outcomes are 5^3 = 125 Thus, Probability = 60125 = 1225 Therefore, the correct option is A.

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