What Are Combinations? Part 1
In math, sometimes order matters, and sometimes it doesn’t. When it comes to permutations and combinations, it matters a lot. Keep on reading to find out why.
Jason Marshall, PhD
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What Are Combinations? Part 1
How many ways can the colors of a rainbow be arranged? As we found out last time, the answer to this question depends upon what we mean by “arranged.”
For example, do we count a rainbow that has red, orange, yellow, green, blue, and purple stripes from top to bottom (like a normal rainbow) as the same as a rainbow with this sequence of stripes in the exact opposite order (like the upper of the two rainbows in a double rainbow)? Or should we consider these two to be entirely different? It depends upon whether you’re looking for the number of permutations or the number of combinations.
In the last episode we learned how to calculate permutations, which means that today we’re going to learn about calculating combinations. So, how do you do it? That’s exactly the question we’ll be tackling today.
Recap: Calculating Permutations
As we learned last time, to find the number of permutations of a group of objects we have to count up the number of ways the objects can be arranged into different orders. For example, a three-digit combination lock in which each digit can take on one of ten values—like 1-2-3, 5-1-4, 9-9-2, etc.—has 10 x 10 x 10 = 1,000 possible permutations.
The number of permutations depends upon whether or not the objects you’re arranging can repeat.
The number of permutations depends upon whether or not the objects you’re arranging can repeat. In a normal combination lock, the numbers can repeat (since a code like 1-1-1 is perfectly acceptable). On the other hand, if the digits aren’t allowed to repeat (which would be weird but possible), the total number of permutations would be 10x9x8=720.
If you’re not sure how I got these numbers, you might want to check out last week’s show before continuing on … because we’re going to be using some of these ideas today.
From Permutations to Combinations
Instead of a lock where the numbers have to be entered in the correct order, let’s imagine a strange lock in which the order doesn’t matter—where the lock opens any time the right three numbers are entered … no matter what order they’re entered in. For a lock like this, the different permutations of the correct combination of numbers is irrelevant.
If we want to know how many different codes such a lock could have, we don’t need to count the number of permutations of its three numbers (since many of those will be duplicates of each other as far as the lock is concerned). Instead, we need to count the number of so-called combinations. In other words, we need to ignore the ordering of the numbers and just count the number of possible groupings.
How to Calculate Combinations Without Repeats
Getting back to our many-striped rainbow: How many color combinations are possible for a given number of stripes? In other words, how many groupings of stripes are possible when we completely ignore the ordering of the stripes? If you think about it, you’ll see that the answer to this question depends upon whether or not the colors of stripes are allowed to repeat. Can we have a rainbow with five red stripes and one blue stripe? Or do all of the stripes have to be different colors? Let’s take these cases one at a time.
Up first, how can we find the number of stripe combinations in a rainbow whose colors are not allowed to repeat? The trick to calculating this kind of combination without repeats is to:
- Start by pretending that you’re not calculating combinations at all, and instead calculate the number of permutations (just like before).
- Once you have the number of permutations, divide it by the number of possible arrangements of the items in a permutation to get the number of combinations.
Combinations in Imaginary Rainbows
To see how this works, let’s think about a simplified rainbow with only three stripes. Using what we learned last time, we can figure out that this rainbow has 6x5x4=120 possible permutations of six colors into three stripes (with no repeat colors). But we want to know the number of combinations, which means we don’t care about the order of the colors: we just care about a particular grouping of colors.
There are 20 possible color combinations in a three-striped rainbow.
To go from permutations to combinations, we need to know the number of ways that any group of three colors can be arranged. If you think about it, you’ll see that there are three options for the first stripe, then two options for the second (since one option has already been selected), and one option for the third stripe. So, there are a total of 3x2x1=6 options. This means that there are 120/6=20 possible color combinations in a three-striped rainbow. Make sense?
Going back to our original six-striped rainbow, how many possible combinations does it have when each of its six stripes must have a unique color? As before, we first need to calculate the number of permutations that are possible for the six colors of a six-striped rainbow. As we learned to calculate last time, there are 6! = 720 possible permutations.
How many ways can these six different colors be arranged? Well, there are 6 choices for the first stripe, 5 for the second, 4 for third, 3 for the fourth, 2 for the fifth, and 1 for the last stripe. In other words, there are 6! = 720 different arrangements for these six colors. So, the number of combinations is 720 / 720 = 1. Wait … what?! If you think about it, this isn’t such a surprise. There are six colors and six stripes, so although there are many possible permutations, there’s only one combination containing all the colors.
Combinations in the Real World
All this talk of counting color combinations of rainbows might seem kind of contrived, and if you’re thinking that, you’re right—this isn’t something you’ll probably ever have to worry about in your life. But although the problem might be contrived, the mathematics of combinations definitely isn’t. In fact, it shows up all the time in the real world.
For example, how can you figure out your chances of winning a six-number lottery? You start by figuring out the number of possible combinations of the six numbers! That’s all we’ll say about this for now … we’ll come back and talk about it in more detail soon.
But before we’re done with combinations, we have one more type to learn about. Namely, what if those stripes in our hypothetical rainbow are allowed to repeat? How many different combinations would be possible then? Sadly, as so often happens, we’re all out of time for today. So, the answers to this question will have to wait until next time.
Wrap Up
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Combination lock image from Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.