Question

The number of ways in which 7 letters can be put in 7 envelopes such that exactly four letters are in wrong envelopes is

• A. $300$
• B. $315$
• C. $325$
• D. $1035$

Answer 1

Answer: (B)

$315$

Three letters out of 7 goes in correct envelope and 4 goes in wrong envelope.
So, total number of arrangements ${=}^7{C}_3$
Total number of dearrangements $=4!(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!})$

Hence, number of ways in which exactly 4 goes in wrong envelope

$=\frac{\mathrm{7.6.5}}{\mathrm{3.2.1}}.4!.[\frac{1}{2}-\frac{1}{6}+\frac{1}{24}]$

$=35(12-4+1)==315$