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Hard Combination/Permutation Problem

The Jefferson School District employs 5 full-time teachers and 4 teaching aides. The district requires each classroom to have a teaching team consisting of 1 full-time teacher and 2 teaching aides. How many different teaching teams are possible?

A) 20

B) 24

C) 30

D) 60

E) 120

The worst mistake you can make on a problem like this (or similar problems), is to make up a formula or do some random calculations in your head.

Many times, you might be tempted to do 5 * 4 or do 5+4+3+2+1 or do 5*4*3*2*1 or some other calculation without any understanding of why.

The best way to attack a problem like this is to start writing out all the possible combinations. Once you start writing out the combinations 1) you will see a pattern 2) if you don’t see a pattern, keep writing more till you see a pattern.

So for this problem, let’s abbreviate the 5 full time teachers: A, B, C, D, E Let’s call the 4 aides: 1, 2, 3, 4.

Start writing out your possible combinations. Remember you need to have a team of 1 full-time teacher and 2 aides.

  1. A12
  2. A13
  3. A14
  4. A21 (Remember, I can’t have this because A12 is the same as A21)
  5. A23
  6. A24
  7. A34

Pro-tip: Make sure to work systematically. Don’t jump all over the place when writing out your combinations. See how I went from A12 -> A13 ->A14 -> A23 and so on.

Now I start writing them out with B:

  1. B12
  2. B13
  3. B14
  4. B23
  5. B24
  6. B34

At this point, or maybe even earlier, you should have realized that with each full-time teacher, we can have 6 teams. I have 5 full-time teachers and they can each have 6 teams. Therefore 5 * 6 = 30.

Answer: C

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