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?
$ {\text{A}}{\text{. }}{{\text{7}}^4} \\
  {\text{B}}{\text{. }}{}^7{{\text{P}}_4} \\
  {\text{C}}{\text{. 28}} \\
  {\text{D}}{\text{. 4}} \\ $

Answer 1

HINT: First select 4 cards out of 7 and use the combination formula and then rearrange them to get the answer.

Complete step-by-step solution -
It is given in the question that 7 New year greeting cards are present and these 7 greeting cards need to be distributed amongst 4 friends.
So, firstly select the 4 cards from the 7 cards as I have to give only one card to one friend.
Here, we will apply the concept of choosing n different objects from m different objects and here we have to choose 4 different objects from the given 7 different objects and this can be done by ${}^7{{\text{C}}_4}$ .
As these 4 cards are different and also the friends are different so these 4 cards can be arranged in those number of ways in which we can arrange ‘a’ different things and this can be done by a!.
Here,
$a = 4$
So, this arrangement can be done by 4! ways.
The total number of ways by which these greeting cards can be send will be = ${}^7{{\text{C}}_4} \times 4!$
And the above notation is similar to ${}^7{{\text{P}}_4}$.
The correct option is B.

NOTE: Here instead of using Combination one can use the permutation formula directly where a person has to arrange r objects out of n given items.