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Question

# In a party, there are 10 married couples. Each person shake hands with every persion other than her or his spouce. The total numberof hand shakes exchanged in that party is ______?

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Solution

## 10 maried couple means total 20 person First we will calculate total handshake than we will subtract handshake between husband and her wife. Total handshake between 20 person will be 20c2 (Detail proof of this i attached in the bottom) Total 20 person and 2 person will do handshake Total handshake = 20*19*18!/(18!)*2! Total handshake = 20*19/2 = 190 Handshake between husband and wife = 10 So Total number of handshake exchanged in that party = 190 - 10 =180 Answer : 180 If there are two people at a party, they can shake hands once. There is no one else left to shake hands with, so there is only one handshake total. 2 people, 1 handshake If there are three people at a party, the first person can shake hands with the two other people (two handshakes). Person two has already shaken hands with person one, but he can still shake hands with person three (one handshake). Person three has shaken hands with both of them, so the handshakes are finished. 2 + 1 = 3. 3 people, 3 handshakes If there are four people at a party, person one can shake hands with three people, person two can shake hands with two new people, and person three can shake hands with one person. 3 + 2 + 1 = 6. 4 people, 6 handshakes Are you seeing a pattern? If you have five people, person five shakes four other hands, person four shakes three other hands, person three shakes two other hands, and person two shakes one hand. Another way to see it is, Person 5 Person 4 Person 3 Person 2 4 + 3 + 2 + 1 = 10 handshakes total ==================================================== People at Party Number of Handshakes 2 1 3 1 + 2 = 3 4 1 + 2 + 3 = 6 5 1 + 2 + 3 + 4 = 10 6 1 + 2 + 3 + 4 + 5 = 15 . . . n 1 + 2 + ...+ (n-1) = n(n-1)/2 So for 20 person 20*19/2 = 190

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