Q. 63.8( 25 Votes )

# 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.

Answer :

Let *L*1, *L*2, *L*3 be three letters and *E*1, *E*2, and *E*3 be their corresponding envelops respectively.

Let L_{i}E_{i} denote i^{th} letter is inserted in i^{th} envelope

∴ Sample space is

L_{1}E_{1}, L_{2}E_{3}, L_{3}E_{2},

L_{2}E_{2}, L_{1}E_{3}, L_{3}E_{1},

L_{3}E_{3}, L_{1}E_{2}, L_{2}E_{1},

L_{1}E_{1}, L_{2}E_{2}, L_{3}E_{3},

L_{1}E_{2}, L_{2}E_{3}, L_{3}E_{1},

L_{1}E_{3}, L_{2}E_{1}, L_{3}E_{2},

∴ 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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