Six three-representative delegations attend an international conference. The representatives shake hands when they are introduced to one another. How many handshakes are possible if each delegate shakes hands only once with every other attendant except with those of his/her delegation?

A) 36
B) 72
C) 90
D) 135
E) 388
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IMO D.

There are 6 delegation, each contains 3 representatives.

Consider 1st delegation:
Member 1 from delegation 1 will handshake with every other person in other 5 delegations. So total handshakes = no of people in other delegations except 1st = 15.
Similarly logic for other two members of 1st delegation.
So, total handshakes from delegation 1 = 15*3 = 45.

Delegation 2:
For members in delegation 2, consider delegation members other than 1 & 2 . Remaining members - 12 (4 delegations)
Total handshakes - 12*3 = 36.

So applying same logic
Total handshakes = 15*3 + 12*3 + 9*3 + 6*3 + 3*3 = 45+36+27+18+9 = 135.

Tip: No need to consider handshakes of last delegation. Since the same has been already covered in previous cases.

I think answer should be D.
Total number of members: 6 * 3 = 18
Total number of handshakes(including among same delegations) : 18C2 = 153
Now, Total handshakes among same delegations = 6 * 3C2 = 18
So, Answer = 153 - 18 = 135

We are given that there are 3 representatives from 6 different delegations. So, there are a total of 18 representatives.

If every representative were to shake hands with all other representatives (meaning all 18 reps would shake hands), this would happen in the following number of ways:

18C2 = (18 x 17)/2! = = 9 x 17 = 153 ways

However, since each person shook hands with every person not from his or her own delegation, we can subtract out the number of times those handshakes occurred.

Since each company has 3 reps, the number ways those three reps can shake hand is 3C2 = (3 x 2)/2! = 3 ways, and since there are 6 companies, this would occur 6 x 3 = 18 times.

Thus, the number of ways for the reps to shake hands with every person not from his or her own delegation is 153 - 18 = 135 ways.

Alternate Solution:

Let’s count the total number of handshakes. Since each of the 6 x 3 = 18 representatives shake hands with 5 x 3 = 15 other representatives, this makes 18 x 15 = 270 handshakes. However, we counted each handshake twice, once for each of the parties involved in the handshake. Therefore, the number of handshakes that take place is 270 / 2 = 135.

Answer: D
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