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# In how many different ways can a group of 8 people be

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In how many different ways can a group of 8 people be [#permalink]

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12 Jan 2012, 06:59
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In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?

A. 90
B. 105
C. 168
D. 420
E. 2520
[Reveal] Spoiler: OA

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Re: In how many different ways [#permalink]

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12 Jan 2012, 07:06
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manalq8 wrote:
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?

90
105
168
420
2520

$$\frac{C^2_8*C^2_6*C^2_4*C^2_2}{4!}=105$$, we are dividing by 4! (factorial of the # of teams) as the order of the teams does not matter. If 8 people are - 1, 2, 3, 4, 5, 6, 7, 8, then (1,2)(3,4)(5,6)(7,8) would be the same 4 teams as (5,6)(7,8)(1,2)(3,4), as we don't have team #1, team #2...

For the first person we can pick a pair in 7 ways;
For the second one in 5 ways (as two are already chosen);
For the third one in 3 ways (as 4 people are already chosen);
For the fourth one there is only one left.

So we have 7*5*3*1=105

You can check similar problems:
probability-88685.html?hilit=different%20items%20divided%20equally
probability-85993.html?highlight=divide+groups
combination-55369.html#p690842
sub-committee-86346.html?highlight=divide+groups

There is also direct formula for this:

1. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is not important is $$\frac{(mn)!}{(n!)^m*m!}$$.

2. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is important is $$\frac{(mn)!}{(n!)^m}$$

Hope it helps.
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Re: In how many different ways can a group of 8 people be [#permalink]

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01 Mar 2013, 06:07
For the first person we can pick a pair in 7 ways;
For the second one in 5 ways (as two are already chosen);
For the third one in 3 ways (as 4 people are already chosen);
For the fourth one there is only one left.

Hi Bunuel,

Can u tell y have u considered 7 ways above?

Also,in the previous method im not clear why we need to divide by 4!.
Is it like say for eg:how many ways can we arrange the word AEEB so we need to consider 4!/2! since EE is repeated?

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Re: In how many different ways can a group of 8 people be [#permalink]

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01 Mar 2013, 20:09

Suppose you were choosing a person to pair with person 1. You could form the following pairs
(1,2),(1,3)(1,4),(1,5),(1,6),(1,7)(1,8)
That's a total of 7 choices possible. Hence, one needs to choose 7.

Hope this helps!

shreerajp99 wrote:
For the first person we can pick a pair in 7 ways;
For the second one in 5 ways (as two are already chosen);
For the third one in 3 ways (as 4 people are already chosen);
For the fourth one there is only one left.

Hi Bunuel,

Can u tell y have u considered 7 ways above?

Also,in the previous method im not clear why we need to divide by 4!.
Is it like say for eg:how many ways can we arrange the word AEEB so we need to consider 4!/2! since EE is repeated?

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Re: In how many different ways can a group of 8 people be [#permalink]

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01 Mar 2013, 21:05
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shreerajp99 wrote:
For the first person we can pick a pair in 7 ways;
For the second one in 5 ways (as two are already chosen);
For the third one in 3 ways (as 4 people are already chosen);
For the fourth one there is only one left.

Hi Bunuel,

Can u tell y have u considered 7 ways above?

Also,in the previous method im not clear why we need to divide by 4!.
Is it like say for eg:how many ways can we arrange the word AEEB so we need to consider 4!/2! since EE is repeated?

Let me try explain what I thought:
Person 1 to pair with someone - have 7 choices (out of 8)
Person 2 to pair with somone - have 5 choices (out of remaining 6 people, note the 2nd person is also included in remaining 6)
.......Likewise

Divide by 4!, you are close to correct, it is to avoid repeats of similar groups. Since order of the the chosen groups does not matter here (Person 1, Person 2) is same as (Person 2, Person 1) - that means as per the formula we have number of groups which includes these repeats, to negate those we divide by 4! to get a realistic number with no such repeats

I tried, hope it is clear.....

Thanks
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Re: In how many different ways can a group of 8 people be [#permalink]

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03 Mar 2013, 05:31
Hmm Bunuel something I did quickly, and forgot the context to do this in (this doesn't happen as frequently) was to select by team? How would I approach the problem then. IE instad of 8C2, I started with 4C1 and started to proceed from there, with the hopes of multiplying by 2! to account for the different arrangements we could have within each team (but not by 4! to account for the order of these different teams)

Why would 4C1 not be appropriate in this case? Is it because those 4 teams aren't set beforehand?

I am messing this up conceptually and want to correct this mistake
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Re: In how many different ways can a group of 8 people be [#permalink]

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03 Mar 2013, 23:02
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manimgoindowndown wrote:
Hmm Bunuel something I did quickly, and forgot the context to do this in (this doesn't happen as frequently) was to select by team? How would I approach the problem then. IE instad of 8C2, I started with 4C1 and started to proceed from there, with the hopes of multiplying by 2! to account for the different arrangements we could have within each team (but not by 4! to account for the order of these different teams)

Why would 4C1 not be appropriate in this case? Is it because those 4 teams aren't set beforehand?

I am messing this up conceptually and want to correct this mistake

Check out this post: http://www.veritasprep.com/blog/2011/11 ... ke-groups/

It discusses two different questions:
1. Distributing 12 different chocolates equally among 4 boys (similar to splitting 8 people in 4 distinct teams with 2 people each - your question)
2. Distributing 12 different chocolates equally in 4 stacks (similar to splitting 8 people in 4 teams of 2 people each - the original question)

See if grouping makes sense thereafter.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 18153 [2], given: 236 Manager Joined: 07 Feb 2011 Posts: 105 Kudos [?]: 64 [0], given: 45 Re: In how many different ways can a group of 8 people be [#permalink] ### Show Tags 05 Mar 2013, 18:02 VeritasPrepKarishma wrote: manimgoindowndown wrote: Hmm Bunuel something I did quickly, and forgot the context to do this in (this doesn't happen as frequently) was to select by team? How would I approach the problem then. IE instad of 8C2, I started with 4C1 and started to proceed from there, with the hopes of multiplying by 2! to account for the different arrangements we could have within each team (but not by 4! to account for the order of these different teams) Why would 4C1 not be appropriate in this case? Is it because those 4 teams aren't set beforehand? I am messing this up conceptually and want to correct this mistake Check out this post: http://www.veritasprep.com/blog/2011/11 ... ke-groups/ It discusses two different questions: 1. Distributing 12 different chocolates equally among 4 boys (similar to splitting 8 people in 4 distinct teams with 2 people each - your question) 2. Distributing 12 different chocolates equally in 4 stacks (similar to splitting 8 people in 4 teams of 2 people each - the original question) See if grouping makes sense thereafter. That example and wording was extremely confusing and frustrating. I am still trying to see what language prompted the difference in permutation vs combination. Maybe my brain has been fried this week (lots of pracatice, one full length CAT), but the big thing in this problem is you DO NOT account for order TWICE the 4! is for the team and the 2! is within every single combination for each pair _________________ We appreciate your kudos' Kudos [?]: 64 [0], given: 45 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7799 Kudos [?]: 18153 [0], given: 236 Location: Pune, India Re: In how many different ways can a group of 8 people be [#permalink] ### Show Tags 05 Mar 2013, 19:09 manimgoindowndown wrote: That example and wording was extremely confusing and frustrating. I am still trying to see what language prompted the difference in permutation vs combination. Maybe my brain has been fried this week (lots of pracatice, one full length CAT), but the big thing in this problem is you DO NOT account for order TWICE the 4! is for the team and the 2! is within every single combination for each pair Yes, because in this question, the groups are not distinct. You have to split 8 people in 4 groups. You can split them like this: (A, B), (C, D), (E, F), (G, H) or like this: (G, H), (A, B), (C, D), (E, F) they are the same split. They are not assigned to group1, group2, group3, group4. Say the total number of ways we get = N Now, if we change the question and say that we have 8 people and we need to divide them into 4 teams: Team 1, Team 2, Team 3 and Team 4 Then, one split is this: Team 1 = (A, B); Team 2 = (C, D), Team 3 = (E, F), Team 4 = (G, H) and another split is: Team 1 = (G, H), Team 2 = (A, B); Team 3 = (C, D), Team 4 = (E, F) These two cases were the same in our original question but if the teams/groups are distinct, the two cases are distinct. Now, the total number of ways = N*4! _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: In how many different ways can a group of 8 people be [#permalink]

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22 Jun 2014, 08:27
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In how many different ways can a group of 8 people be [#permalink]

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23 Jun 2014, 10:56
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manalq8 wrote:
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?

A. 90
B. 105
C. 168
D. 420
E. 2520

You need to choose 4 groups of 2 from 8 people: That's $$\frac{8!}{2!2!2!2!}$$, then divided by $$4!$$, since any ordering of these 4 groups is the same.

Total = $$\frac{8!}{2!2!2!2!4!} = \frac{8!}{8*2*4!} = \frac{7!}{2*4!} = \frac{7*6*5*4*3*2}{4*3*2*2} = \frac{7*6*5}{2} = 35*3 = 105.$$

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Re: In how many different ways can a group of 8 people be [#permalink]

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30 Nov 2015, 05:12
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Re: In how many different ways can a group of 8 people be [#permalink]

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17 Jul 2017, 03:54
manalq8 wrote:
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?

A. 90
B. 105
C. 168
D. 420
E. 2520

Let's select the 4 groups of 2 individuals each

8C2 * 6C2 * 4C2 * 2C2

But wait... There is some problem...

Let 8 individuals are
A B C D E F G H

Case 1: selected 4 groups of 2 are respectively

A&B C&D E&F G&H

i.e. A&B us first group, C&D is second group etc

Case 2: selected 4 groups of 2 are respectively

C&D A&B G&H E&F respectively

So case 1 and 2 are the same except that order of the groups has been accounted for

The arrangements of 4 groups can be done in 4! Which needs to be excluded from calculation

Hence divide the result by 4!

So final result = 8C2 * 6C2 * 4C2 * 2C2 / 4! = 105

Hope this helps!!!
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Re: In how many different ways can a group of 8 people be [#permalink]

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30 Aug 2017, 06:07
Hi all,

here you wants to divide 8 people in group of 4 teams in 2 members .
So you can do these directly

8!/2!*2!*2!*2!*4!

so in these in the numerator denotes the total number of people and four times 2! denotes the four group and 4! denotes as the all the group has same number.

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Re: In how many different ways can a group of 8 people be [#permalink]

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30 Aug 2017, 06:22
Bunuel wrote:
manalq8 wrote:
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?

90
105
168
420
2520

$$\frac{C^2_8*C^2_6*C^2_4*C^2_2}{4!}=105$$, we are dividing by 4! (factorial of the # of teams) as the order of the teams does not matter. If 8 people are - 1, 2, 3, 4, 5, 6, 7, 8, then (1,2)(3,4)(5,6)(7,8) would be the same 4 teams as (5,6)(7,8)(1,2)(3,4), as we don't have team #1, team #2...

For the first person we can pick a pair in 7 ways;
For the second one in 5 ways (as two are already chosen);
For the third one in 3 ways (as 4 people are already chosen);
For the fourth one there is only one left.

So we have 7*5*3*1=105

You can check similar problems:
http://gmatclub.com/forum/probability-8 ... %20equally
http://gmatclub.com/forum/probability-8 ... ide+groups
http://gmatclub.com/forum/combination-5 ... ml#p690842
http://gmatclub.com/forum/sub-committee ... ide+groups

There is also direct formula for this:

1. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is not important is $$\frac{(mn)!}{(n!)^m*m!}$$.

2. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is important is $$\frac{(mn)!}{(n!)^m}$$

Hope it helps.

How did you use the formula? I did not get the answer using formula.

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In how many different ways can a group of 8 people be [#permalink]

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30 Aug 2017, 09:45
Expert's post
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gayatriv21 wrote:
Bunuel wrote:
manalq8 wrote:
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?

90
105
168
420
2520

$$\frac{C^2_8*C^2_6*C^2_4*C^2_2}{4!}=105$$, we are dividing by 4! (factorial of the # of teams) as the order of the teams does not matter. If 8 people are - 1, 2, 3, 4, 5, 6, 7, 8, then (1,2)(3,4)(5,6)(7,8) would be the same 4 teams as (5,6)(7,8)(1,2)(3,4), as we don't have team #1, team #2...

For the first person we can pick a pair in 7 ways;
For the second one in 5 ways (as two are already chosen);
For the third one in 3 ways (as 4 people are already chosen);
For the fourth one there is only one left.

So we have 7*5*3*1=105

You can check similar problems:
http://gmatclub.com/forum/probability-8 ... %20equally
http://gmatclub.com/forum/probability-8 ... ide+groups
http://gmatclub.com/forum/combination-5 ... ml#p690842
http://gmatclub.com/forum/sub-committee ... ide+groups

There is also direct formula for this:

1. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is not important is $$\frac{(mn)!}{(n!)^m*m!}$$.

2. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is important is $$\frac{(mn)!}{(n!)^m}$$

Hope it helps.

How did you use the formula? I did not get the answer using formula.

We are dividing mn = 8 people into m = 4 teams of n = 2 people each.

Use the first formula, sine the order does not matter: $$\frac{(mn)!}{(n!)^m*m!}=\frac{8!}{(2!)^4*4!}=105$$.

P.S. You don;t really to memorize this formula for the GMAT.
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