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.

(A) 2/3

(B) 5/3

(C) 7/3

(D) None of these

Answer 1

Step 1:

Let the envelope be denoted by E₁, E₂, E₃ and the corresponding letters by L₁, L₂, L₃

At least one letter should be in right envelope.

Let us consider all the favorable outcomes

Step 2:

(i) 1 letter in correct envelope and 2 in wrong envelope.

(ie) (E₁L₁,E₂L₃,E₃L₂),(E₁L₃,E₂L₂,E₃L₁),(E₁L₂,E₂L₁,E₃L₃)

(ii) Two letter in correct envelope.

(ie) (E₁L₁,E₂L₂,E₃L₃)

∴ No of favorable outcomes =4

Step 3:

Total no of outcomes =3!=3×2×1=6

∴ Required probability =n(E)/n(S)

⇒4/6

⇒2/3

Hence (A) is the correct answer.