Statistics can be fairly straightforward — when you know what you're looking for. That's why it's important to always remember the difference between terms. This is especially true when it comes to permutations and combinations. Here are two key ways to keep things in order.
Some ACT questions will ask you how many ways there are to put something in order. The trick here is to remember that what you think of as a combination lock — a sequence where the order of numbers matters — is actually what the math world (and the ACT) would call a permutation lock.
Either way, you'll want to draw a dash for each space you need to fill and then write the number of possibilities in each one. After doing that, figure out if the order matters — words like "arrangement" or "different" may signal this. If it does, then you're solving a permutation, and you can simply multiply the numbers in the dashes together. If the order does not matter (watch for words like "group"), then it's a combination and after you multiply the numbers in the dashes you must then also divide by the factorial of the number of spaces.
It's easier to solve an ordering problem once you draw it out, so we'll also write out the steps for these next two examples.
Example 1: Anthony, Bedda, Cathy, Dinesh and Earl are running for student government. One will be president, one will be vice president, and one will be treasurer. How many different outcomes are there?
Step 1: Fill in your dashes and multiply.
There are three positions to be filled, so write down three dashes. Dash 1 has five possible candidates who can occupy the role. Dash 2 will have four, and Dash 3 will have three. Now multiply. Written out, this looks like:
5 x 4 x 3
Step 2: Determine if it's a permutation or combination.
We know this is a permutation because order matters. Once the president role is taken, the student in that position cannot also be vice president. Additionally, the word "different" is a hint that directs us to permutation.
Step 3: Solve.
Because the order matters for permutations, stop here and solve. In this example, there are 60 arrangements or permutations possible.
But what if, instead of having different roles, three of the five students are being chosen to serve on a committee? In that case, you need to cancel the duplicates. What are duplicates? Well, if Bedda is president, Dinesh is VP, and Earl is treasurer, that's a different order than Earl as president, Bedda as VP, and Dinesh as treasurer. But if those three people are just on a committee together (in a single group), then order doesn't matter and you must divide by the number of ways to arrange the three of them (3 x 2 x 1). Let's look at how you would solve this type of problem.
Example 2: Anthony, Bedda, Cathy, Dinesh, and Earl are running for student government, but only three can be selected to join the committee. How many different groups of students can make up the committee?
Your first two steps in solving a combination are exactly the same as solving a permutation. Keep on keeping on! That leaves you with 5 x 4 x 3 = 60. Now, however, things get different.
Step 4: Divide and solve.
Because order doesn't matter, we need to divide by 3!, and just to be clear, that exclamation point is a factorial symbol, not an expression of our excitement. A factorial requires you to multiply the given number by each subsequent integer down to 1, so in this case, 3 x 2 x 1, or 6. This means that your 60 possible arrangements will be divided by 6. Therefore, we know there are 10 groups or combinations that can make up the student government committee.
There's a big difference between the 60 outcomes from the permutation and the 10 outcomes from the combination. Always remember it, because the ACT will likely offer both choices on a question of this nature, hoping that you'll confuse the two.
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