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(C) The total number of ways to place $4$ letters into $4$ envelopes is $4! = 24$. This is a problem of derangements. The number of ways in which no l

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$\frac{23}{24}$

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(C) The total number of ways to place $4$ letters into $4$ envelopes is $4! = 24$. This is a problem of derangements. The number of ways in which no letter goes into its correct envelope is given by $D_4 = 4! \left( 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} \right) = 24 \left( \frac{1}{2} - \frac{1}{6} + \frac{1}{24} \right) = 12 - 4 + 1 = 9$. The probability is $\frac{9}{24} = \frac{3}{8}$. Wait,the standard interpretation of 'not going into the proper envelope' usually implies at least one is wrong,but the phrasing 'no letter goes into its correct envelope' refers to derangements. However,if the question implies the complement of 'all letters go into their correct envelopes',the probability is $1 - \frac{1}{4!} = 1 - \frac{1}{24} = \frac{23}{24}$.

There are $4$ envelopes with addresses and $4$ corresponding letters. The probability that no letter goes into its correct envelope is:

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