The forumla for combinations and thinking in larger chunks

I hate the formula for combinations. Here it is:

For n objects, the number of different ways to choose k of them is: n! / (n-k)! k!

For example, if I have 25 students and need to choose a committee of 5 of them, there are:
25! / 20! 5! possible committees.

The reason I hate the formula is because it gives virtually no clue about where it comes from or how it makes sense. Students forget it and misapply it because it doesn't fundamentally make any sense.

I always think of this kind of problem in this way: 25*24*23*22*21 / 5!

The numerator connects very straightforwardly to other counting problems. The 25*24*23 etc. represent how many students I could choose as the 1st student for my committee, the 2nd student for my committee, the 3rd student, etc. I have to divide because I'm over-counting the number of committees that are really different. For example, the committee with students {A, B, C, D, E} is really the same as the committee with students {B, C, A, D, E}--the order that I choose the students in doesn't matter. How much to divide by? Each committee I might choose is the same as 5! -1 other committees, because that's how many ways to re-order the same 5 people.

This is probably too quick an explanation to feel comfortable unless you've worked with other combinatorics problems recently--but my students have taken to it quite well after understanding the more fundamental types of counting it relies on.

This seemed to me a great leap forward in student understanding. I didn't tell them that there was a "formula for combinations" because I feared that they would be unwilling to just think about how the problem made sense. As I said, this worked out very well until I had to give them more elaborate problems:

How many 5-card poker hands have (exactly) 2 aces and 2 kings? This is a trickier problem to think through from first principals. It's a fairly long process with many subtle places to go wrong. It would be much easier to think about it in terms of a few independent choices:

(# ways to choose 2 aces)*(# ways to choose 2 kings)*(# of ways to choose another card), or

(4 choose 2)*(4 choose2)*(43 choose 1)

This is where the abstraction of (n choose k) would be handy. It would let students chunk the problem more easily.

This reminded me of something I read somewhere about quick thinkers. The author compared the mind to a pipe with water (thoughts) flowing through it. A lot of people think that fast thinkers are like pipes with faster running water; the thoughts come sequentially at a faster rate. This might happen sometimes, but far more common is for a quick thinker to have a wider pipe--they use abstractions so their thoughts simply contain more, even if they happen at the same speed as everyone elses.

The use of combinations and permutations seems to me a good example of this. A student /could/ reason their way, step by step to the correct solution, but that's many sequential steps. If they can think of the problem in terms of combinations and permuatations, each of their fundamental operations encapsulates several of the old sequential steps at once. This lets their minds arrive at the same place faster, and with less chance of error (or wandering, or boredom).

In my class, students who have an ability to focus their attention for longer periods didn't have trouble with the longer problems. It was students who "got lost" in the steps due to lapses of attention, or an inability to conceptualize the process as a whole, that had trouble.

Next time I'm going to try and achieve a best-of-both-worlds. I'll teach permuations and combinations my way, but then make sure they recognize each as a fundamental type of situation; and help them make the gestalt switch to breaking larger problems down into them, instead of building them from the ground up, as they're used to.