December 2014

# Solution

This is how I have solved it. Let me know whether it is right.

Let there be "n" number of people. Each person shakes hands with three others. Then, there should be a total of 3n handshakes. This, in fact, is not right. It is double the total number of handshakes that actually take place. Why?

Let me take a smaller number of people to answer why. Let there be four persons - P1, P2, P3 and P4; let each of them shake hands with three others. So, there should be 3n or 3 x 4 = 12 handshakes. Let's see if this is true.

Let's represent the handshakes as ordered pairs. What I mean is: If P1 shakes hands with P2, I represent it as (P1, P2).

So, P1 shakes hands as follows: (P1, P2), (P1, P3) and (P1, P4).

P2 shakes hands as follows: (P2, P1), (P2, P3) and (P2, P4).

P3 shakes hands as follows: (P3, P1), (P3, P2) and (P3, P4).

P4 shakes hands as follows: (P4, P1), (P4, P2) and (P4, P3).

Counting the number of ordered pairs I get 12 as the number of handshakes. But (P1, P2) and (P2, P1) are one and the same because P1 shaking hands with P2, and P2 shaking hands with P1 is the same. I have, therefore, to consider it only once. So is the case with (P1, P3) and (P3, P1); (P1, P4) and (P4, P1); (P3, P2) and (P2, P3); (P2, P4) and (P4, P2); and (P3, P4) and (P4, P3). Therefore, I have to consider them only once. Thus, the number of handshakes is, in fact, 12/2 = 6.

Similarly, if there are "n" people and each person shakes hands with exactly three others, the number of handshakes will be 3n/2.

Now, we have been told that there are 24 handshakes in all.

Therefore, 3n/2 = 24.

So, 3n = 48. Thus n = 48/3 = 16. Therefore, there are 16 people.

I was wondering if there were 17 people, would it have been possible for each person to shake hands with exactly three others. I don't think it is possible. Suppose there are 17 people and each person shakes hands exactly with three others, then the total number of handshakes will be (17 x 3)/2 = 25.5. How can there be 25.5 handshakes? With 17 people, I don't think the condition that each should shake hands with exactly three others can be fulfilled. What do you say?

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