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Re: If there are four distinct pairs of brothers and sisters [#permalink]
28 Aug 2010, 16:32

3

This post received KUDOS

Expert's post

seekmba wrote:

If there are four distinct pairs of brothers and sisters, then in how many ways can a committee of 3 be formed and NOT have siblings in it?

8 24 32 56 192

I find it difficult to understand the difference between permutation and combination and hence find these questions very hard.

As committee shouldn't have siblings in it, then a pair can send only one "representative" to the committee. # of ways to choose which 3 pairs of brothers and sisters should send one "representative" to the committee is C^3_4 (choosing 3 pairs which will be granted the right to send one "representative" to the committee);

But each of these 3 pairs can send 2 persons to the committee either a brother or a sister: 2*2*2=2^3;

Re: If there are four distinct pairs of brothers and sisters [#permalink]
06 Sep 2010, 11:29

The first person on the committee can be anyone of the 8.

The second person on the committee can be only one out of 6 (the first person with her or his sibling excluded).

The third person can be selected only out of 4 (first two members and their siblings excluded), giving 8*6*4 permutations. These are called permutations because we count every distinct combination of 3 people 6 times (6=3!). For example, we count {Ann, Bill, and Jane} as {Ann, Bill, Jane}, {Ann, Jane, Bill}, {Jane, Ann, Bill}, {Jane, Bill, Ann}, {Bill, Ann, Jane}, {Bill, Jane, Ann}. To get the number of combinations, divide the number of permutations by 6: 8*6*4 / (6) = 32 combinations.

I understand uptil the last part but I don't understand why we're diving by 6? Can anyone clarify this for me?

Re: If there are four distinct pairs of brothers and sisters [#permalink]
06 Sep 2010, 12:10

5

This post received KUDOS

Expert's post

SnehaC wrote:

The first person on the committee can be anyone of the 8.

The second person on the committee can be only one out of 6 (the first person with her or his sibling excluded).

The third person can be selected only out of 4 (first two members and their siblings excluded), giving 8*6*4 permutations. These are called permutations because we count every distinct combination of 3 people 6 times (6=3!). For example, we count {Ann, Bill, and Jane} as {Ann, Bill, Jane}, {Ann, Jane, Bill}, {Jane, Ann, Bill}, {Jane, Bill, Ann}, {Bill, Ann, Jane}, {Bill, Jane, Ann}. To get the number of combinations, divide the number of permutations by 6: 8*6*4 / (6) = 32 combinations.

I understand uptil the last part but I don't understand why we're diving by 6? Can anyone clarify this for me?

We need to divide 8*6*4=192 by the factorial of the # of people - 3! to get rid of duplications 8*6*4=192 contains ---> \frac{192}{3!}=32 - correct answer.

Consider example with smaller number: there are two couples and we want to choose 2 people not married to each other. Couples: A_1, A_2 and B_1, B_2. Committees possible:

A_1,B_1; A_1,B_2; A_2,B_1; A_2,B_2.

Only 4 such committees are possible.

If we do as proposed in the solution you posted: The first person on the committee can be anyone of the 4. The second person on the committee can be only one out of 2 (the first person with her or his sibling excluded).

So we'll get: 4*2=8, so more than 4, which means that 8 contains some duplications. And to get the right answer we should divide 8 by 2! --> 8/2!=4.

It's because if you pick A1 for the first pick and than pick B1 for the second you'll get the committee {A1, B1} but if you'll pick B1 for the first pick and then A1 you'll get the exact same committee {A1, B1} (dividing by the factorial of the # of people in committee you'll exclude this double countings).

Re: If there are four distinct pairs of brothers and sisters [#permalink]
06 Sep 2010, 17:54

Bunuel, i like the way u use quick formulas for permutations and combinations..can u let me know how can i get these formulas? i want to strengthen my skills on this subject..

Re: If there are four distinct pairs of brothers and sisters [#permalink]
07 Sep 2010, 03:11

Expert's post

vdixit wrote:

Bunuel, i like the way u use quick formulas for permutations and combinations..can u let me know how can i get these formulas? i want to strengthen my skills on this subject..

Thanks!

Probability and Combinatorics chapters of Math Book:

Re: If there are four distinct pairs of brothers and sisters [#permalink]
27 Dec 2012, 18:38

How many ways to select 3 of the pairs with representative in the group from 4 pairs? 4!/3!1! = 4 How many ways to select a representative from each pair? 2 x 2 x 2 = 8 4*8 = 32

Re: If there are four distinct pairs of brothers and sisters [#permalink]
11 Jul 2014, 22:42

we can select 8 people for 1st place, 6 for second (only one from pair can be selected) and , 4 for 3rd So we can have total = 8*6*4 = 192 Now in above calculation, we have counted all no of ways. (ABC is different from ABE) so we have to divide the above value with no of ways we can select 3 people

No. of ways to select 3 people = 3!

so 192/3! = 32

gmatclubot

Re: If there are four distinct pairs of brothers and sisters
[#permalink]
11 Jul 2014, 22:42