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by Patrick_GMATFix » Sat Feb 22, 2014 7:28 pm
Think of each handshake as a selection of 2 people from the 10 people in the room. If we assume that everyone shook hands with every one else, the number of handshakes corresponds to the number of ways to pick 2 from 10. The combination formula applies since order doesn't matter (John shaking Suzy's hand & Suzy shaking John's hand = same handshake)

Combination formula is: number of ways to pick r items from n available = n!/[(r!)(n-r)!]

Solution: 10!/(2!8!) = 45 handshakes

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by [email protected] » Sat Feb 22, 2014 7:31 pm
Hi YavuzSelim111,

This question can be solved in a couple of ways: a high-concept math approach or a "brute-force" answer that anyone can use. I'll focus on the second method.

Since we have 10 people, who will all shake hands with one another, we know that each pair of people will lead to 1 hand shake (and a person CAN'T shake hands with himself or herself).

If we call the people ABCDE FGHIJ

Person A will shake hands with BCDE FGHIJ = 9 shakes

Person B ALREADY shook hands with A, so we won't shake hands again....
Person B will shake hands with CDE FGHIJ = 8 shakes

Person C ALREADY shook hands with A and B, so we won't shake hands again....
Person C will shake hands with DE FGHIJ = 7 shakes

Notice the pattern 9, 8, 7.....the numbers will shrink by 1 with every letter, so we'll end up with...

9+8+7+6+5+4+3+2+1+0 = 45 total handshakes.

Final Answer: C

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by Brent@GMATPrepNow » Sat Feb 22, 2014 10:00 pm
YavuzSelim111 wrote:At the end of a banquet 10 people shake hands with each other. How many handsakes will there be total?

A. 100
B 20
C 45
D 50
E 90
Another approach:

Once everyone has shaken hands, ask each of the 10 people, "How many people did you shake hands with?"
We'll find that EACH PERSON shook hands with 9 people, which gives us a total of 90 handshakes (since 10 x 9 = 90).

From here we need to recognize that every handshake has been counted TWICE. For example, if Person A and Person B shake hands, then Person A counts it as a handshake, AND Person B also counts it as a handshake. Of course only one handshake occurred.

To account for the duplication, we'll divide 90 by 2 to get 45

Answer: C

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