# Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.

Let L1, L2, L3 be three letters and E1, E2, and E3 be their corresponding envelops respectively.

Let LiEi denote ith letter is inserted in ith envelope

Sample space is

L1E1, L2E3, L3E2,

L2E2, L1E3, L3E1,

L3E3, L1E2, L2E1,

L1E1, L2E2, L3E3,

L1E2, L2E3, L3E1,

L1E3, L2E1, L3E2,

there are 6 ways of inserting 3 letters in 3 envelops.

And there are 4 ways in which at least one letter is inserted in proper envelope. (first 4 rows of sample space)

Probability that at least one letter is inserted in proper envelope =

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