Q59 / 155 · WBJEE 2019 · Mathematics
There are 7 greeting cards, each of a different colour and 7 envelopes of same 7 colours as that of the cards. The number of ways in which the cards can be put in envelopes, so that exactly 4 of the cards go into envelopes of respective colour is,
- A${ }^{7} C_{3}$
- Correct answerB$2^{7} C_{3}$
- C$3!^{4} C_{4}$
- D$3!^{7} C_{3}^{4} C_{3}$
Solution
Required number of ways
$$
\begin{array}{l}
={ }^{7} C_{4} \times D(3) \\
={ }^{7} C_{3} \times\left[3 \cdot\left(1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}\right)\right] \\
{\left[\because{ }^{n} C_{r}={ }^{n} C_{n-r, a n d}\right.} \\
\left.D(n)=n\left[1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\ldots\right]\right] \\
={ }^{7} C_{3} \times 2=2\left({ }^{7} C_{3}\right)
\end{array}
$$
Hence, the correct answer is option: 'B'.