A person writes letters to 6 friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes? Notation : D n = n ! ∑ i = 0 n - 1 i i !

Question

A person writes letters to 6 friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes? Notation : D n = n ! &#8721; i = 0 n - 1 i i !

Aicharya · Accepted Answer

The number of ways in which at least two of the letters are in the wrong envelopes is given by
$\sum_{r = 2}^{6} {}^{6}C_{6 - r} D_{r} = {}^{6}C_{4} D_{2} + {}^{6}C_{3} D_{3} + {}^{6}C_{2} D_{4} + {}^{6}C_{1} D_{5} + {}^{6}C_{0} D_{6}$
where $D_{r} = r! (\sum_{i = 0}^{r} \frac{{(- 1)}^{i}}{i!})$

A person writes letters to 6 friends and addresses the corresponding envelopes. In how many ways can the…

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