After a typist has written ten letters and had addressed the ten corresponding envelopes, a careless mailing clerk inserted the letters in the envelopes at random, one letter per envelope. What is the probability that [b]exactly[/b] nine letters were inserted in the proper envelopes?
1. Let's denote the letters as \( L_1, L_2, \ldots, L_{10} \) and the corresponding envelopes as \( E_1, E_2, \ldots, E_{10} \).
2. We are asked to find the probability that exactly nine letters are inserted into their corresponding envelopes.
3. If nine letters are correctly placed in their corresponding envelopes, then the tenth letter must also be correctly placed in its corresponding envelope because there is only one envelope left.
4. This means that if nine letters are correctly placed, the tenth letter must also be correctly placed, resulting in all ten letters being correctly placed.
5. Therefore, it is impossible to have exactly nine letters correctly placed without the tenth letter also being correctly placed.
6. Hence, the probability of exactly nine letters being correctly placed is zero.

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