Question

A person writes letters to six 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 wring envelope.

Answer 1

The no. of ways in which at least two of them are in wrong envelopes

${\sum }_{r-2}^n={}^n{C}_n-rDr$

$={}^n{C}_n-2D2{+}^n{C}_n-3D3+{}^n{C}_n-4D4{+}^n{C}_n-5D5{+}^n{C}_n-6D6$

${=}^6{C}_4.2!(1-\frac{1}{1!}+\frac{1}{2!})+{}^6{C}_3.3!(1-\frac{1}{1!}\frac{+1}{2!}\frac{-1}{3!}+\frac{1}{4!}\frac{-1}{5!})$

$+{}^6{C}_6.6!(1-\frac{1}{1!}+\frac{1}{2!}\frac{-1}{3!}+\frac{1}{41}-\frac{-1}{5!}+\frac{1}{6!})$

$=15+40+135+264+265$

$719$