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Problem Solving - Combinations

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Problem Solving - Combinations

by sparkles3144 » Thu Apr 25, 2013 1:52 pm
There are 7 people at a dinner. Before sitting down, each person shakes hands with everybody else. How many handshakes were there?

I actually did it using addition... 6+5+4+3+2+1 = 21

Using Combinations it is something like 7C2
Why is it 7C2 and not 7C1?

Thank you so much!
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by Ben.Miller@ApexGMAT » Thu Apr 25, 2013 2:41 pm
7C2 denotes a pair of, in this case, people. In math, we call this a pair as well, or when order matters, an ordered pair.

So we get:

1 shaking hands with 2-7 (6 total)
2 shaking hands with 3-7 (5 total)
and so on until 6 shaking hands with 7

7C1 means choose a SINGLE member from the group. Hence, there are 7 possibilities (person 1, 2, 3,... 7).
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by Brent@GMATPrepNow » Thu Apr 25, 2013 4:04 pm
sparkles3144 wrote:There are 7 people at a dinner. Before sitting down, each person shakes hands with everybody else. How many handshakes were there?

I actually did it using addition... 6+5+4+3+2+1 = 21

Using Combinations it is something like 7C2
Why is it 7C2 and not 7C1?

Thank you so much!
Ben Miller is absolutely right in that 7C2 represents the # of ways to select 2 people (to shake hands).

Here's another approach that doesn't even use formal counting techniques.

After each person has shaken hands with everyone else, we could ask each person "How many people did you shake hands with?"

Each of the 7 people will have shaken hands with 6 other people.
So, the total number of handshakes appears to be 7x6 = 42

Of course, this number is not correct, because we have counted each single handshake two times (A says he shook hands with B, AND B says he shook hands with A).

So, to handle the duplication, we must divide 42 by 2 to get 21.

Cheers,
Brent

PS: If anyone is interested, we have a free video on calculating combinations (like 7C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789
Brent Hanneson - Creator of GMATPrepNow.com
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