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Question
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.
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Solution
Let the envelopes be denoted as A, B, C and the corresponding letters as a, b, c respectively.
(i) Ways of putting one letter in its proper envelope and the other two in the wrong envelopes
(Aa, Bc, Cb), (Ac, Bb, Ca), (Ab, Ba, Cc)
(ii) If two letters are put in the proper (correct) envelopes, then the third will also be in the proper (correct) envelope.
(iii) There is a way to put all the three letters in their proper (correct) envelopes (Aa, Bb, Cc).
Ways of putting letters in at least one proper envelope
3 + 1
= 4
Total ways of putting three letters in three envelopes = 3! = 6
∴ Probability of putting at least one letter in proper envelope = `4/6 = 2/3`
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