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 to divide by 2?
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Why, or rather what says to divide by 2 at the end?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
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
12, 13, 14, 15, 16, 17
23, 24, 25, 26, 27
34, 35, 36, 37
45, 46, 47
56, 57
67
21 pairs
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!/(72)! = 7! / 5!
If order matters not(our case), we have to divide by 2 (pair)
7! / ( (72)! * 2) = 7! /(5! * 2) = 7*6 / 2 = 21
We have to eliminate those redundancies. How many of those? 2.
The equation is (if order matters)
7!/(72)! = 7! / 5!
If order matters not(our case), we have to divide by 2 (pair)
7! / ( (72)! * 2) = 7! /(5! * 2) = 7*6 / 2 = 21

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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
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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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 nonmathematical terms  if order doesn't matter, count up up the number of items in your numerator, call that number "n", and divide by n!.
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 nonmathematical 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