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A clerk has seven letters, each for a different person, and seven addressed envelopes…

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A clerk has seven letters, each for a different person, and seven addressed envelopes. The clerk puts one letter into each envelope at random. Find the number of ways in which no letter is in its own envelope. Find the probability that no letter is in its own envelope, and the probability that at least one is. Then find the probability that exactly five letters are in their own envelopes. Give the probabilities to four places.

Need a reminder? The key ideaShow

All different: $\dfrac{^NP_n}{N^n}$ for $n$ objects in $N$ equally likely boxes. For matching, $D_n = n!\sum_{k=0}^{n}\dfrac{(-1)^k}{k!}$ counts the arrangements with no match, and exactly $r$ matches have probability $\dfrac{^nC_r \times D_{n-r}}{n!}$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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