These concepts are important for solving counting problems on the ACT® Math section. Let's break them down and see how they're used in real ACT® questions.
Factorial
Factorial is this exclamation point in math! Many students haven’t seen this before so they don’t know what it is.
6! (pronounced "6 factorial") means multiply 6 by every whole number less than it, down to 1.
So
Think of it like a countdown multiplication. Start with the number, then multiply by each number below it until you hit 1.
You'll rarely see questions that are just about calculating factorials. Usually, the ACT® Math asks you to understand what factorial means and then use it in a slightly more complex problem.
Permutation
There's a specific formula for permutations, but knowing the concept is often more useful than memorizing the formula. The ACT® typically focuses on understanding rather than formula recall.
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A permutation is all about arranging things in order. The important phrase to remember is "order matters."
Let's look at an example:
A debate team has six members. How many different ways can a first, second, and third place winner be assigned to the debate members?
In this case, order matters. The person in first place can't simultaneously be in second or third. We need to find the number of ways to arrange 6 people, taking 3 positions at a time.
= = = 120
So there are 120 different ways to assign first, second, and third place among six team members.
Combination
Combination is about selecting groups where the order doesn't matter. The key phrase here is "order doesn't matter."
The formula for combination is:
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This might look similar to the permutation formula, but the extra in the denominator accounts for the fact that we don't care about the order of selection.
Let's look at an example:
A committee of 5 people is to be formed from a group of 10 people. How many ways can this committee be formed?
In this case, we don't care about the order in which people are selected for the committee. Whether Alice is picked first or last, it's still the same committee.
= = = 252
So there are 252 different ways to form a committee of 5 people from a group of 10.
You'll see several permutation, combination and factorial questions on the test. Ask yourself: Does the order matter in this situation? That will help you decide whether to use permutation or combination.