GMAT Math

bpolley00 wrote:... I mean I know you aren't assuming anything, there must be a way to distingusih this. How did you make that distinction as it is vitally important on the test - if I see one of these God awful question I want to get it right.

Selections and arrangements are different things.
When you have to arrange something, first you have to select them.

For example, say there are 5 people but 3 chairs. In how many ways the 5 people can sit on the 3 chairs?

First we have to select 3 people from 5 people ---> Selection
In how many ways we can do that? ---> 5C3 = 5!/[(2!)*(3!)] = 10

Now, for each of the above selection, we have to arrange this 3 people in three chairs ---> Arrangement
In how many ways we can do that? ---> 3! = 6 ways

hence, total number of arrangements = [5!/[(2!)*(3!)]]*(3!) = 5!/2! = 60

In mathematical terminology,

• Selection == Combination (Formula --> nCr = n!/[((n - r)!)*(r!)]
  Arrangement = Permutation (Formula --> nPr = n!/[(n - r)!]

So the trick is to understand whether the problem is asking for number of selections or arrangements? If it is asking for number of selections only, we stop after selection. But if it is asking for number of arrangements, first we will select then arrange.

For example, the question you chose,

bpolley00 wrote: A fraternity must choose a delegation of 3 senior members and 2 junior members for an annual conference. It has 12 senior and 11 junior candidates, how many different delegations are possible?

It is obvious that we need to select 3 from 12 seniors and 2 from 11 juniors. But after that do we need to arrange them? No. Why not? Because this is a delegation, a simple team forming. For example, if A, B, and C are three seniors and X and Y are two juniors, ABCXY/CABYX/BXACY etc all are same team. Therefore, we don't need number of arrangements. Calculating number of selections is sufficient.

Hence, number of different delegations = (Number of ways to select 3 from 12)*(Number of ways to select 2 from 11) = (12C3)*(11C2)

But what if the question said that in a delegation there will be a post of representative or leader and only a senior can occupy that place?

Now, in a delegation there will be 3 seniors and 2 juniors with one senior occupying the post of leader. This means for each selection we made earlier, the delegation will change depending upon which one of the three seniors is the leader.

If we assume the first position is for leader, then ACBXY and CABXY are different delegations. But ACBXY/ACXBY/ABCYX etc are still same delegation as the other posts are inter-changeable.

Now, number of delegations = (Number of ways to select 3 from 12)*(Number of ways to select 2 from 11)*(Number of ways to arrange 3 seniors among themselves) = (12C3)*(11C2)*(3!) = (12P3)*(11C2)