General - Combinatorics Question

[mayank]

Hi,

This questions is not specific to the Strategy guides but is a general question about combinatorics, I wasn't sure where to post it since the GeneralMath forum has been closed.

I need some help understanding a Combinatorics concept. I have been using the anagram method and the combinations formula.

I want to understand how to calculate the maximum possible combinations (order does not matter) for questions like:

[Mr. and Mrs. X want to have 4 children...] - At this point how do I calculate all the possible combinations of Boy and Girls they can have if they have 4 children.

One way is the Slot method,

2 (B/G) x 2 (B/G) x 2 (B/G) x 2 (B/G) = 64 (for each slot there are 2 options B or G) but I would like to understand how to get to this result from the other approaches mentioned above.

But if I were to say, they want exactly 2 Boys and 2 Girls then the slot method fails (or alteast the way I using it).

2 (B/G) x 1 (B) x 2 (B/G) x 1 (G) = 4? It should be 6.

Another type of question is (very similar to the one above):

A committee of 2 has to be selected from a pool made up of male and female members out of 10 candidates (5 male, 5 female). Again same issue how do I calculate the maximum number of possibilities.

Aka, I having trouble grasping the idea where out of a pool there are two types of candidates and we have to make combinations for all of them.

Can you please walk me through how I would use slot, combinations formula and/or anagram for these?

Thanks for your help.

[tim]

Hi Mayank,
i'm going to go ahead and claim that there is little if any difference among the methods you describe. In fact, if you do it right, there is no difference even permutations and combinations. Basically the approach you want to take is to treat each independent choice as a process of breaking a large group into smaller groups. Put the large group on top with a factorial and all the small groups on the bottom with factorials. Of course the numbers on the bottom must add up to the number on top. This approach completely replaces both the combinations and permutations formulas, and the toughest part of any such question is to determine what your large groups and small groups are. Then just multiply all your independent choices together. Let's apply this process to your questions:

"[Mr. and Mrs. X want to have 4 children...] - At this point how do I calculate all the possible combinations of Boy and Girls they can have if they have 4 children."

Each child is an independent choice; for each choice you have two genders to choose from, so you choose one gender to assign to the child and one gender not to assign. Large group is thus 2 and small groups are 1 and 1, so you have 2!/(1!1!)=2. Of course this is WAY more complicated than just noting the obvious fact that there are two choices. :) Multiplying the four independent choices gives us 2*2*2*2=16..

"But if I were to say, they want exactly 2 Boys and 2 Girls then the slot method fails (or alteast the way I using it)."

This time our choices are not independent, and instead of taking a child and choosing a gender we are taking 4 children and choosing 2 to be boys and 2 to be girls. The large group is 4 and the small groups are 2 and 2. 4!/(2!2!)=6..

"A committee of 2 has to be selected from a pool made up of male and female members out of 10 candidates (5 male, 5 female). Again same issue how do I calculate the maximum number of possibilities."

For this one, you have not specified whether we must have a male and a female or it doesn't matter. i'll do the problem both ways to demonstrate the distinction:

If genders don't matter, you just have 10 people from which you must choose 2 to be on the committee and 8 not to be. Large group is 10, small groups are 2 and 8. 10!/(2!8!)=45..

If we need one of each gender, we make two independent choices. First we need a male; from a group of 5, we select 1 and leave 4 unselected. 5!/(1!4!)=5. The calculation for the female is exactly the same and again we get 5. Multiply those and we get 25..

As a bonus, let's turn this into a probability problem: given the scenario you described, what is the probability that the committee consists of a male and a female? 25 is the number of successful outcomes, and there are 45 total ways two people can be selected. This gives us 25/45 or 5/9. Of course, we can also get 5/9 another way: no matter who is chosen first, the second person is chosen from 9 people, and 5 of those 9 are of the opposite gender.. :)

Whenever you're doing combinatorics or probability problems for practice, be sure to do them as many different ways as you can. That way you'll have several options to choose from when you are faced with a similar problem on the GMAT..

i hope this helps..