# Number of handshakes - exclusion apporach

5 indian and 5 american couples meet at a party and shake hands. If no wife shakes hands with her husband and no indian wife shakes hands with a male, then the number of hand shakes that take place at the party is?

I know this question has been asked before here and I understand the solution but I can't figure out what am I missing in my approach.

Assumption - Everyone shakes hands with everyone they can according to these restrictions, and that no one shakes hands with the same person twice.

Here's what I did -

1. Total number of handshakes by Indian females: ${10 \choose 2}$ since they can only shake hands with the american wives.
2. The total number of handshakes by the other fifteen people will be: ${15 \choose 2}$
3. Subtract from that number of handshakes by american wives with their own husband: 5

So the answer should be ${10 \choose 2}$ + ${15 \choose 2}$ - 5 = 145

But the answer is 135. Can somebody point out what I am I missing in this approach?

• You're missing the assumptions that everyone shakes hands with everyone they can according to these restrictions, and that no one shakes hands with the same person twice. You made those assumptions, but you never said that you did. Also, the answer in the post you linked is much larger than $135$. Where did that come from? – Arthur Nov 24 '17 at 6:45
• @Arthur not sure I follow. The indian wives can only do 10C2 handshakes and for the rest of the 15 people, you can make groups of 2 in 15C2 ways. – aandis Nov 24 '17 at 6:49
• @Arthur in the linked question, there are 10 couples each and not 5. Hence the number is higher. I understand that the assumptions may not be obvious. I'll edit the question and add those. – aandis Nov 24 '17 at 6:52
• I missed the fact that your question is about five couples and the question you link is about twenty couples. At any rate, your $\binom{10}2$ which you say only counts handshakes done by Indian wives also includes the ten handshakes between the American wives. – Arthur Nov 24 '17 at 6:54
• @Arthur good catch. Please add this as an answer and I'll be happy to accept. – aandis Nov 24 '17 at 6:58

When you calculate the number of handshakes done by the Indian wives, $\binom{10}2$ also picks up the handshakes done between all the American wives. There are ten of those.