Question

A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes, is
(a) $\frac{1}{4}$

(b) $\frac{11}{24}$

(c) $\frac{15}{24}$

(d) $\frac{23}{24}$

Answer 1

$\left(\mathrm{d}\right)\frac{23}{24}$

4 letters can be placed in 4 envelopes in 4! ways = 24 ways
Now, there is only one method, by which all the letters are placed in the right envelope.

P(all letters are placed in the right envelopes) = $\frac{1}{24}$

P(all letters are not placed in the right envelopes) = 1 $-$ P(all letters are placed in the right envelopes)

$$\begin{gathered} =1-\frac{1}{24} \\ =\frac{23}{24} \end{gathered}$$