different possible senior delegations. (With practice, you'll be able to produce that kind of calculation quickly.)Sometimes, the GRE will combine smaller combinatorics problems into one larger problem. In this case, you'll need to deal with successive or multiple arrangements.
Fortunately, at the end of the day, you do the same thing you've been doing all along: multiply!
If a GRE problem requires you to choose two or more sets of items from separate pools, count the arrangements separately. Then multiply the numbers of possibilities for each step.
Distinguish these problems—which require choices from separate pools—from complex problems that are still single arrangements (all items chosen from the same pool), for which you create a single set of slots.
For instance, say you have to choose 1 treasurer, 1 secretary, and 3 more representatives from one class of 20 students. This might seem like two or more separate problems, but it requires just one set of slots. Set up 5 slots, labeled T, S, R, R, and R.
| _____ | _____ | _____ |
_____ |
_____ |
||||
| T | S | R |
R |
R |
Fill in the slots. You can pick from all 20 students for the first slot (it doesn't matter that you're picking the treasurer first), 19 for the second slot, and so on.
| __20__ | __19__ | __18__ |
__17__ |
__16__ |
||||
| T | S | R |
R |
R |
Finally, you multiply those numbers together and divide by 3!, the factorial of the number of repeated labels.
20 × 19 × 18 × 17 × 16 = 1,860,480.
Dividing by 3! = 6, you get 310,080.
But let's now look at a different scenario:
The I Eta Pi fraternity must choose a delegation of 3 senior members and 2 junior members for an annual interfraternity conference. If I Eta Pi has 12 senior members and 11 junior members, how many different delegations are possible?
This problem involves two genuinely different arrangements: 3 seniors chosen from a pool of 12 seniors, and 2 juniors chosen from a separate pool of 11 juniors. These arrangements should be calculated separately.
First, choose the 3 senior delegates from a pool of 12 seniors. You'll label the 3 slots the same, so you'll get
different possible senior delegations. (With practice, you'll be able to produce that kind of calculation quickly.)
Next, choose the 2 junior delegates from a pool of 11 juniors. Again, you'll label the 2 slots the same, so you'll get
different possible junior delegations.
Finally, because the choices are successive and independent, you multiply the results you got from each little sub-problem.
220 × 55 = 12,100 different delegations are possible.
For each of the 220 senior delegations, you have all 55 different junior delegations available. This brings us all the way back to the sandwich problem at the start of this chapter. For each of the 2 choices of bread, you had all 3 fillings available. So you multiply the 2 and the 3 to get 6 possible sandwiches. Ultimately, multiple arrangements work the same way.
Three men (out of 7) and 3 women (out of 6) will be chosen to serve on a committee. In how many ways can the committee be formed?