Q70 / 120 · VITEEE 2012 · Mathematics
There are 5 letters and 5 different envelopes. The number of ways in which all the letters can be put in wrong envelope, is
- A119
- Correct answerB44
- C59
- D40
Solution
Required numbers
$$
=5 !\left[1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\frac{1}{5 !}\right]=44
$$
Note that if $r(0 \leq r \leq n)$ objects occupy the original places and none of the remaining $(\mathrm{n}-\mathrm{r})$ objects occupies its original places then the number of such arrangements
$$
\begin{array}{l}
={ }^{n} C_{r} \cdot(n-r) ! \\
{\left[1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\ldots .+(-1)^{n-2} \frac{1}{(n-r) !}\right]}
\end{array}
$$
Hence, the correct answer is option: 'B'.