Question

If you have $7$ new year greeting cards and you want to send them to $4$ of your friends, in how many ways can this be done?

• A. $7^4$
• B. ${}^7{P}_4$
• C. $28$
• D. $4$

Answer 1

The correct option is B ${}^7{P}_4$

If you don't want to use direct permutation then follow this solution.

You have to do 2 things -

(i) Select 4 cards from the 7 cards (as you have to give only one card to your friend) - It is basically choosing m objects from n different objects.

Here $m=4,n=7$, so it can be done in ${}^7{\text{C}}_4$ways.

(ii) Arrange these 4 cards among the friends as cards are different and so are the friends - it is like arranging k different things in a line which can be done in $k!$ ways, here $k=4$ so it can be done in $4!$ ways .

Now the total ways in which greeting cards can be send are $={}^7{\text{C}}_4\times 4!=$ ${}^7{\text{P}}_4$ ways.