Why to divide by 2?

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white: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Tue Dec 23, 2008 12:05 pm

by white » Wed Apr 29, 2009 11:31 pm

How many different pairs can be selected from a group of 7 people?

The answer is 21?

I just can't understand why to divide at the 2 at th end?!

Thank you

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Source: Beat The GMAT — Problem Solving | Wed Apr 29, 2009 11:31 pm

marcusking: Master | Next Rank: 500 Posts; Posts: 106; Joined: Wed Feb 04, 2009 6:09 am; Location: Louisville, KY; Thanked: 8 times

Re: Why to divide by 2?

by marcusking » Thu Apr 30, 2009 5:16 am

white wrote:How many different pairs can be selected from a group of 7 people?

The answer is 21?

I just can't understand why to divide at the 2 at th end?!

Thank you

Why, or rather what says to divide by 2 at the end?

Just write out the numbers 1 to 7 and start pairing them together It should be quick enough and you should see a patern develop and probably won't need to even write it all the way out.

Possible pairs
1-2, 1-3, 1-4, 1-5, 1-6, 1-7
2-3, 2-4, 2-5, 2-6, 2-7
3-4, 3-5, 3-6, 3-7
4-5, 4-6, 4-7
5-6, 5-7
6-7

21 pairs

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g2000: Junior | Next Rank: 30 Posts; Posts: 19; Joined: Tue Jan 09, 2007 5:50 am; Thanked: 1 times

combination problem

by g2000 » Thu Apr 30, 2009 6:55 am

Given 7 people(A,B...G), we don't care if A is chosen first or last.
We have to eliminate those redundancies. How many of those? 2.

The equation is (if order matters)
7!/(7-2)! = 7! / 5!

If order matters not(our case), we have to divide by 2 (pair)
7! / ( (7-2)! * 2) = 7! /(5! * 2) = 7*6 / 2 = 21

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lilu: Master | Next Rank: 500 Posts; Posts: 148; Joined: Wed Dec 10, 2008 5:13 pm; Location: SF, CA; Thanked: 12 times

by lilu » Thu Apr 30, 2009 9:07 am

We need to divide by 2 because you calculate EACH pair by this formula:
7!/5!=42
If you'd have a set of letters ABCDEFG and AB would be different from BA, then the 42 pairs would be correct. But here a pair of two people does not change depending on whether you pick one person first and the other person second, then will still be a pair. This is why you're counting those pairs twice and need to divide by 2:
7!/5!2!=21

The more you look, the more you see.

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VP_Jim: GMAT Instructor; Posts: 1223; Joined: Thu May 01, 2008 3:29 pmFollowed by: 15 members; Location: Los Angeles, CA; Thanked: 185 times

by VP_Jim » Thu Apr 30, 2009 5:38 pm

Technically, you're dividing by 2! (2 factorial), not 2. I think it's better to think of it as the factorial so that you can answer other, similar questions such as:

How many teams of 3 people can you make out of 7 possible choices?

That would be:

(7x6x5)/(3x2x1) = 35

The way I like to think about it is - in very non-mathematical terms - if order doesn't matter, count up up the number of items in your numerator, call that number "n", and divide by n!.

Jim S. | GMAT Instructor | Veritas Prep

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