Question

There are $5$ letters and $5$ different envelopes. The number of ways in which all the letters can be put in wrong envelope, is

• A. 119
• B. 44
• C. 59
• D. 40

Answer 1

Answer: (B)

Required numbers
$=5 !\left[1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}=\frac{1}{5 !}\right]=44$
Note if $r(0 \leq r \leq n)$ objects occupy the original places and none of the remaining $(n-r)$ objects occupies its original places then the number of such arrangements $={ }^{n} C_{r}(n-r) !$
$\left[1 \frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\ldots+(-1)^{n-2} \frac{1}{(n-r) !}\right]$