If there are 6 letters and 6 corresponding en | vedclass

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(A) The number of ways to place $n$ letters in wrong envelopes is given by the derangement formula $D_n = n! \left[ 1 - \frac{1}{1!} + \frac{1}{2!} -

Vedclass · Accepted Answer

$265$

Vedclass · Answer

(A) The number of ways to place $n$ letters in wrong envelopes is given by the derangement formula $D_n = n! \left[ 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + (-1)^n \frac{1}{n!} \right]$.For $n = 6$,the number of ways is $D_6 = 6! \left[ 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \frac{1}{6!} \right]$.$D_6 = 6! \left[ \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \frac{1}{6!} \right]$.$D_6 = \frac{720}{2} - \frac{720}{6} + \frac{720}{24} - \frac{720}{120} + \frac{720}{720}$.$D_6 = 360 - 120 + 30 - 6 + 1 = 265$.

If there are $6$ letters and $6$ corresponding envelopes,in how many ways can all the letters be placed in the wrong envelopes?

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