Three letters, to each of which corresponds an envelope, are placed in the envelopes at random. The probability that all the letters are not placed in the right envelopes, is

Question

Three letters, to each of which corresponds an envelope, are placed in the envelopes at random. The probability that all the letters are not placed in the right envelopes, is (A)  (1/6)  (B)  (5/6)  (C)  (1/3)  (D)  (2/3)

Tardigrade Admin · Accepted Answer

Three letters can be placed in 3 envelopes in

3!

ways, whereas there is only one way of placing them in their right envelopes.
So, probability that all the letters are placed in the right envelopes

= \frac{1}{3 !}

Hence, required probability

= 1 - \frac{1}{3 !} = \frac{5}{6}

Q. Three letters, to each of which corresponds an envelope, are placed in the envelopes at random. The probability that all the letters are not placed in the right envelopes, is

$\frac{1}{6}$

$\frac{5}{6}$

$\frac{1}{3}$

$\frac{2}{3}$

Three letters can be placed in 3 envelopes in $3!$ ways, whereas there is only one way of placing them in their right envelopes.
So, probability that all the letters are placed in the right envelopes $= \frac{1}{3 !}$
Hence, required probability $= 1 - \frac{1}{3 !} = \frac{5}{6}$