Question

There are $7$ greetings 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. ${}^7{C}_3$
• B. $2\cdot {}^7{C}_3$
• C. $3!{}^4{C}_4$
• D. $3!{}^7{C}_3{}^4{C}_3$

Answer 1

The correct option is B $2\cdot {}^7{C}_3$

First we choose any $4$ cards that will go in their respective envelopes. The number of ways of doing that is ${}^7{C}_4.$

For each of these, we have to arrange the remaining $3$ cards such that none of them are in their right envelopes. That means we require dearrangement of $3$ things which can be done in $3!(\frac{1}{2!}-\frac{1}{3!})=2$ ways

So, total number of ways${}^7{C}_4\times 2$ which is same as ${}^7{C}_3\times 2$