Question

A man has 5 friends. In how many ways can he invite one or more of them to a tea party?
$
  (a){\text{ 30}} \\
  (b){\text{ 31}} \\
  (c){\text{ 32}} \\
  (d){\text{ 25}} \\
$

Answer 1

Hint – In this question compute the total number of ways of inviting or not inviting all the 5 friends using the concept that there are only two possibilities with a single friend that is either he will be invited or he will not be invited. Then find the ways of not inviting any one of them and that will be one only. Using this, find the ways of inviting one or more than one of them.

Complete step-by-step answer:
There are 5 friends of a man.
Now we have to calculate the number of ways he can invite one or more of them to a tea party.
Now as we see that the number of ways he cannot invite any of them = 1 (as there are only one possible way of not inviting any of them).
Now there are only two possible ways of inviting a man either invited or not invited.
So the total number of ways of either inviting or not inviting 5 friends = $2 \times 2 \times 2 \times 2 \times 2 = 32$
So the total number of ways of inviting 5 friends = total number of ways of either inviting or not inviting 5 friends – number of ways of not inviting any of them.
So the total number of ways of inviting 5 friends = 32 – 1 = 31.
So this is the required answer.
Hence option (B) is the correct answer.

Note – In this question there was nothing related to arrangements thus we have not used the concept of permutation otherwise for example if this would have been a question related to the sitting arrangement of these 5 friends also then we would have used $5!$ permutations as well.