A group of five people are all working on the same mathematics problem. On the night before it is due, they call each other to discuss their work. Each person talks to all the other people at least once. What is the fewest number of telephone calls that could be made?

Minimum number of calls = 10

Step-by-step explanation:

Lets name the five people as A, B, C, D and E.

On the night before, each person talks to every other person atleast once that means A would talk to B, C, D and E atleast once.

Lets start with A. He would talk to other 4 people which means there would be 4 phone calls made.

Now lets take B. He can talk to A, C, D and E. But A has already talked to C therefore to get minimum number of phone calls, B need not call A again. So he calls only C, D and E.

In case of C using similar logic he need to talk to only D and E.

For D, he talks to E alone.

E does not have to talk to anyone as he has already talked to everyone atleast once.

Total calls = 4 + 3 + 2 + 1

= 10