Question

There are 4 different letters and 4 addressed envelopes. In how many ways can the letters be put in the envelopes so that at least one letter goes to the correct address?

• A. 15
• B. 16
• C. 18
• D. 12

Answer 1

The correct option is A 15

The number of ways in which none of the n letters goes to the correct place
$=n![1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+(-1{)}^n\frac{1}{n!}]$
$\therefore$ Number of ways in which all the 4 letters go to the wrong place
$=4![1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}]=24[1-\frac{1}{1}+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}]=24\left[\frac{12-4+1}{24}\right]=9$
Now the number of ways in which all the possibilities can occur = 4! = 24.
$\therefore$ Number of ways in which atleast one letter goes to the correct place = 24 - 9 = 15.