Guest Post by Chris Seymour
Some math problems on the SAT ask you to count the number of possible ways to do something. Now, you might be thinking, “Come on, Chris, everyone already knows how to count!” But counting problems on the SAT are not as simple as counting on your fingers and toes. Take the following example question:
In order to solve this particular kind of problem, we'll have to tackle two concepts, permutations and combinations. Sound intimidating? You're not alone. Permutation and combination problems are a unique sort of problem on the SAT. To make matters worse, most math classes will tell you to solve these problems using exotic-looking formulas, such as:
The most important difference between permutations and combinations is that order matters for a permutation, while order does not matter for a combination. To understand what this means, let's look at the ice cream problem again.
Remember the rule: Order matters for a permutation, but in a combination order does not matter. So, if we want to determine what kind of problem our ice cream question is, we must consider whether order matters. In other words, does choosing your flavors of ice cream in a different order affect the final outcome? To test this, let's suppose that as your three flavors, you choose chocolate, then strawberry, and then mint chip, in that order.
Next, try choosing the same three flavors in a different order: mint chip, then chocolate, then strawberry. Will the result be different?
The answer is no. All of the flavors are getting mixed together in a bowl, so you get the same mashup of three flavors no matter which flavor goes into the bowl first. The two outcomes are indistinguishable. Since order does not matter, this is a combination problem.
Okay, now we know what a combination is. So what's a permutation? To understand permutations, consider this slightly different problem:
Notice that we still have five flavors of ice cream, and we still want to choose three of those flavors. So, what makes this problem different from the last one? This time, order matters. Switching the order of your choices will cause different children to receive different flavors of ice cream. To confirm this, let's apply the same test as before. First, let's try choosing vanilla, then strawberry, and then cookie dough (in that order).
Now, let's switch the order again. Do we get a different result from choosing cookie dough, then vanilla, then strawberry?
This time, the answer is yes. In the first order we chose, Alice receives vanilla, Bob receives strawberry, and Carol receives cookie dough. But in the second order we chose, Alice receives cookie dough, Bob receives vanilla, and Carol receives strawberry. Even though you are choosing the same three flavors of ice cream, changing the order means different kids end up with different flavors. Since order matters, this is a permutation problem.
So, determining whether you have a permutation or a combination in front of you is simple. Try making all of your choices in one order, then take the same choices and try them in a different order to see if it affects the outcome. If the outcome is the same, then the order doesn't matter, and the problem is a combination. If the outcome is different, order does matter, and the problem is a permutation.
Now that we know how to tell the difference between permutation and combination problems, we can learn how to solve each properly. It turns out that we can use a nearly identical, three-step process to solve any counting problem, including permutation and combination problems. We'll cover this in detail in Part 2.