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.

Question

Philoid · Accepted Answer

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

Let L_iE_i denote i^th letter is inserted in i^th 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 =