Permutation and Combination

[Shib]

Source:IMS GMAT

A photographer will arrange 6 people of 6 different heights for photograph by placing
them in two rows of three so that each person in the first row is standing in front of
someone in the second row. The heights of the people within each row must increase
from left to right, and each person in the second row must be taller than the person
standing in front of him or her. How many such arrangements of the 6 people are
possible?
A. 5
B. 6
C. 9
D. 24
E. 36


To me, if i consider 6 different people with six diff heights and the criteria given in the problem, only one such arrangement is possible.

[JadranLee]

This problem is pretty difficult. Let's assign names to all the positions in the lineup:

BL BM BR
FL FM FR

("B" stands for "back", "F" stands for "front", "L" stands for "left", etc.)

Let's also assign "names" to each of the six people - 1 is the shortest, 2 is the next shortest, ... and 6 is the tallest.

Notice first that the only place where 6 can stand is in the BR position. A person standing in any of the other positions has to be shorter than at least one other person, and 6 isn't shorter than anybody.

By similar reasoning, we can see that the only place where 1 can stand is in the FL position. A person standing in any of the other positions has to be taller than at least one other person, and 1 isn't taller than anybody.

So we know that any possible arrangement will be of this form:

BL BM 6
1 FM FR

All we need to do is count possible ways of putting 2, 3, 4, and 5 in positions BL, BM, FM, and FR. In order to count possibilities, let's focus on who goes into the BL position. 1 and 6 are already fixed in their own positions. There's no way 5 could be in the BL position, because there would be no way to assign someone to BM such that the heights in the back row increased consistently from left to right. So we know that the person in the BL position has to be either 2, 3, or 4. We investigate each possibility in turn:

If 2 goes in the BL position, there are just two possibilities:

2 4 6
1 3 5

and

2 5 6
1 3 4

If 3 goes in the BL position, there are also two possibilities:

3 4 6
1 2 5

and

3 5 6
1 2 4

If 4 goes in the BL position, there is just one possible arrangement:

4 5 6
1 2 3

Counting these possibilities, we see that there are only 5 possible arrangements.

For very difficult combinatorics problems like this one, it is hard to use a formula. Instead, you should draw a picture. Then figure out if there are any positions that are fixed - in this case we saw that 1 had to be at FL, and 6 had to be at BR. Finally, count the remaining possibilities, focusing on one position (in our case we focused on BL) in order to help you organize your thoughts.

[GMAT 2007]

My question would be, is it possible to work through all the different cases/possibilities in 2 min?

GMAT 2007

[StaceyKoprince]

Great question and the answer for most people is probably not.

One of the things you need to study before you take the test is: when do I just let go? The test is trying to find both your upper and lower limits - and it will succeed. You will get questions you can't do in two minutes, no matter how much you study. So when you see a question like this one in practice, ask yourself: at what point should I know that I should just let this go? And is there anything I can do to narrow down my answer choices in some reasonable way to give myself a better shot - an educated guess?

For example, logic tells me that I won't have 36 different possibilities because 6*6 = 36, and there are some significant constraints to "valid" configurations. And that probably means 24 is too high, also. So, I'd guess from among A, B, and C.

Particularly for combinatorics, it's important not to get so sucked in that you lose a ton of time; it's easy to spend 5+ minutes on a problem like this without even realizing it (and probably get it wrong anyway!). Be prepared for that - have an alternative to wasting that time.