There are 4 letters and 4 envelopes. If the l | vedclass

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Question

(D) The number of ways to place $n$ letters into $n$ envelopes such that no letter is in its correct envelope is given by the derangement formula $D_n

Vedclass · Accepted Answer

$3/8$

Vedclass · Answer

(D) The number of ways to place $n$ letters into $n$ envelopes such that no letter is in its correct envelope is given by the derangement formula $D_n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}$.The total number of ways to arrange $4$ letters in $4$ envelopes is $4! = 24$.The number of derangements for $n=4$ is $D_4 = 4! \left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} \right) = 24 \left( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} \right) = 12 - 4 + 1 = 9$.The probability that all letters are in the wrong envelopes is $\frac{D_4}{4!} = \frac{9}{24} = \frac{3}{8}$.

There are $4$ letters and $4$ envelopes. If the letters are placed into the envelopes at random,find the probability that all letters are placed in the wrong envelopes.

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