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In how many ways can a group of 8 be divided into 4 teams of [#permalink]
 22 Dec 2006, 22:37
In how many ways can a group of 8 be divided into 4 teams of 2 people each
1)90
2)105
3) 168
4)420
5) 2520
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answer is 105 (B)
there is probably some formula for this - but i never remember formulas and always get confused in choosing which one to use. so i alwaus count, and use simple reasoning. here is how i did it.
a) how many ways there are to divide 4 people into two group of 2?
that's simple counting: (ab,cd) (ac,bd) (ad,bc) - total 3.
b) how many ways there are to divide 6 people into 3 groups of 2?
there are 5 possibilities for member a:
(ab) (ac) (ad) (ae) (af)
for each such possibility we are left with 4 poeple that need to be grouped into 2 groups of 2. but we already did that (!!!!!!) so for each of the 5 options we have 3 possible completions. total of 5*3=15 options.
c) how many ways to divide 8 people into 4 teams of 2?
use the same reasoning: for person a there are 7 possibilities. and for each such possibility we are eft with dividing 6 people to 3 pairs, which we laready did. hence the answer is 7*(5*3) = 105
with this reasoning (if you followed it fully....) you can answer in no time how many ways there are to divide 12 people into 6 pairs, or 20 people into 10 pairs...
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hobbit wrote: answer is 105 (B)
there is probably some formula for this - but i never remember formulas and always get confused in choosing which one to use. so i alwaus count, and use simple reasoning. here is how i did it.
a) how many ways there are to divide 4 people into two group of 2?
that's simple counting: (ab,cd) (ac,bd) (ad,bc) - total 3.
b) how many ways there are to divide 6 people into 3 groups of 2?
there are 5 possibilities for member a:
(ab) (ac) (ad) (ae) (af)
for each such possibility we are left with 4 poeple that need to be grouped into 2 groups of 2. but we already did that (!!!!!!) so for each of the 5 options we have 3 possible completions. total of 5*3=15 options.
c) how many ways to divide 8 people into 4 teams of 2?
use the same reasoning: for person a there are 7 possibilities. and for each such possibility we are eft with dividing 6 people to 3 pairs, which we laready did. hence the answer is 7*(5*3) = 105
with this reasoning (if you followed it fully....) you can answer in no time how many ways there are to divide 12 people into 6 pairs, or 20 people into 10 pairs...
Boy! what a clear thinking you have !!! Awesome solution 
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Director
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Hi Hobbit,
I am sorry, I am not able to understand your reasoning behind that.
When you say dividing 4 people into 2 groups of 2 each, I got the following # of ways :
Can you please explain ..?
Amit ..
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I got E)
8C2x6C2x4C2x2C2=2520
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BG wrote: I got E)
8C2x6C2x4C2x2C2=2520
I think we need to go one more step ahead.
Consider the individual members as a, b, c, d, e, f, g, and h
Here, the last group 2 C 2 = 1. Say it is g and h.
Now, try to visualize ...
Haven’t we considered this same group while calculating 8 C 2. Yes.
8 C 2 included this group, 6 C 2 included the same group and 4 C 2 did the same thing...
So g-h group was repeated 4 times.
In the same way, the group under 4 C 2 was repeated 3 times.
In the same way, the group under 6 C 2 was repeated 2 times.
In the same way, the group under 8 C 2 was repeated 1 time.
Actual value is = 8C2x6C2x4C2x2C2 / (4 * 3 * 2 *1) = 2520 / 24 = 105
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we can look at it another way....
lets order the8 people and divide into pairs in this way: first and second persons in our ordering are the first pair, third and fourth are another pair and so forth... now if we are going to go over all possible orderings of 8 people we sure going to count all the possible pairings. HOWEVER, there are pairings which are counted more than once. let's se how many times each pairing is counted when going over all possible orderings:
there are total of 8! orderings.
the order in the first pair doesn't matter. so orderings that start with x,y,...
and y,x,... create the same pairing. so we need to divide by 2.
similarly the ordering in the second pair (and the third and the fourth) doesn't matter as well, so we need to divide by 2 for each pair.
in total we have to divide the orderings by 2*2*2*2=16
but that's not the end of it.
the order of the pairs doesn't matter as well. i.e. it doesn't matter if x and y are paired first or second etc...
there are 4!=24 possible ordering of 4 pairs and we need to divide it as well.
so at the end: number of ways = 8!/(2*2*2*2*4!) = 105
this way of counting is also robust, but in my view, is more error prone, as unless you are very experienced and self confident you can always "miscount" - i.e forget to divide something (that is counting the same result more than once), or forget to count a valid solution.
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amit05:
i'm not sure i understood your question.
b.t.w are you an indian "amit" or israeli "amit"?
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Well done, Hobbit, that´s exactly how I did it too (no kidding )
You just use recursive reasoning, starting with the smallest number of items, 2 in this case. Then expand progressively to 4, 6, etc, till you see a pattern or finish counting, whichever happens first.
A formula for these problems would be, given n = number of items (people, etc):
(n-1)*(n-3)*(n-5)...*1
In this case: 7*5*3*1 = 105 (B).
Now on for a beer to cool my brain 
hobbit wrote: answer is 105 (B)
there is probably some formula for this - but i never remember formulas and always get confused in choosing which one to use. so i alwaus count, and use simple reasoning. here is how i did it.
a) how many ways there are to divide 4 people into two group of 2?
that's simple counting: (ab,cd) (ac,bd) (ad,bc) - total 3.
b) how many ways there are to divide 6 people into 3 groups of 2?
there are 5 possibilities for member a:
(ab) (ac) (ad) (ae) (af)
for each such possibility we are left with 4 poeple that need to be grouped into 2 groups of 2. but we already did that (!!!!!!) so for each of the 5 options we have 3 possible completions. total of 5*3=15 options.
c) how many ways to divide 8 people into 4 teams of 2?
use the same reasoning: for person a there are 7 possibilities. and for each such possibility we are eft with dividing 6 people to 3 pairs, which we laready did. hence the answer is 7*(5*3) = 105
with this reasoning (if you followed it fully....) you can answer in no time how many ways there are to divide 12 people into 6 pairs, or 20 people into 10 pairs...
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Andr359 wrote: Well done, Hobbit, that´s exactly how I did it too (no kidding ) You just use recursive reasoning, starting with the smallest number of items, 2 in this case. Then expand progressively to 4, 6, etc, till you see a pattern or finish counting, whichever happens first. A formula for these problems would be, given n = number of items (people, etc): (n-1)*(n-3)*(n-5)...*1 In this case: 7*5*3*1 = 105 (B). Now on for a beer to cool my brain hobbit wrote: answer is 105 (B)
there is probably some formula for this - but i never remember formulas and always get confused in choosing which one to use. so i alwaus count, and use simple reasoning. here is how i did it.
a) how many ways there are to divide 4 people into two group of 2?
that's simple counting: (ab,cd) (ac,bd) (ad,bc) - total 3.
b) how many ways there are to divide 6 people into 3 groups of 2?
there are 5 possibilities for member a:
(ab) (ac) (ad) (ae) (af)
for each such possibility we are left with 4 poeple that need to be grouped into 2 groups of 2. but we already did that (!!!!!!) so for each of the 5 options we have 3 possible completions. total of 5*3=15 options.
c) how many ways to divide 8 people into 4 teams of 2?
use the same reasoning: for person a there are 7 possibilities. and for each such possibility we are eft with dividing 6 people to 3 pairs, which we laready did. hence the answer is 7*(5*3) = 105
with this reasoning (if you followed it fully....) you can answer in no time how many ways there are to divide 12 people into 6 pairs, or 20 people into 10 pairs... Did you just create it or is it actually a formulae
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