Combinations (Makings sets, of a certain size, from another set)
Sometimes you want to form sets of objects from a larger population of objects. For instance, if you are trying to create a team of 5 players, from a set of 10 players who try out, how many different teams can you make? Or, you have 25 t-shirts, but can only pack 5 for a trip. How many different groups of 5 shirts can you make? In these cases, you want to use the combination formula.
The formula for combinations is $\dfrac{n!}{r!(n-r)!}$. If you are allowed to use a calculator, most scientific and graphing calculators also have a combination button, which usually looks like: $\boxed{_nC_r}$.
Before we get into the details of the formula and the button, let's talk about exactly what a combination is.
Combinations take large sets of items and figures out how groups you can make of a smaller number of those objects. For combinations, order doesn't matter. So, if you had the letters ORANGE and you wanted to know how many 3-letter combinations you could make with the letters in ORANGE, the group that includes R, A, and N is one combination (there is no separate group for ANR or NAR or NRA). When you are working with combinations, the order of the items doesn't matter -- just that those items are in a group.
Note: if you care about order, for instance, if, for your purposes, NAR and RAN are different, you want to find the Permutations of the letters ORANGE, not the combinations (see lesson on Permutations for more).
So, clearly, sometimes you want a combination and sometimes you want a permutation. Most generally, if the order of the objects in a group does NOT matter, you want to find combinations. If order does matter, you want to find a permutation.
When might you want to find a combination?
- You walk into a pizza shop. There are 20 toppings. You can only afford 3 toppings. How many different pizzas can you make? Here you want a combination (because, a pizza with pepperoni, olives, and peppers is the same as a pizza with olives, pepperoni, and peppers).
- You are choosing singers for a quartet. If 10 people audition, you might want to see how many different quartets you can make. It doesn't matter which order you hire the singers, a quartet is a quartet, so order does not matter.
- You have a reading list of 50 books to choose from. You're only going to read 5. How many sets of books might you read? As long as you don't care how exactly which book you read first, second, and third, you would want to find how many combinations of book you could read.
Just for clarification, in what cases might you want a permutation?
- You would want to use find the number of permutations, rather than the number of combinations, when the order does matter. If each member of that quartet was going to play a different instrument combinations would tell you how many ways you could group players. It would call these two different combinations: Ben, Jason, Javier, Ashley or Frederick, Jennifer, Yoon, April. Permutations would can these different permutations: Ben on violin, Jason on cello, Javier on base, Ashley on viola or Ben on cello, Jason on viola, Javier on violin, Ashley on bass. Note, these two permutations are the same exact combination! But, because the people are doing different things, it's a different permutation.
Now that you know you want to find the number of possible combinations, how do you find that number?
Whether you use the formula or a calculator, you need to know some pieces of the puzzle.
First make sure that you want a combination (that the order of the objects will not make a difference).
Then figure out the total number of items that are being chosen for the combinations: n
Then figure out the number of items per combination: r
Then you plug those numbers into the formula or into your calculator!
Example: If you are trying to form a musical quartet and 10 students try out for your group, how many different quartets can you make?
Order doesn't matter here, you want to find the number of combinations.
The number of students trying out is 10. $n=10$
The number of students you are choosing for each combination is 4. $r=4$.
Let's try the two different ways of calculating the number of possible combinations:
| Use the formula | Use your calculator |
Important note: ! means factorial. Factorial means to multiply the number times every whole number lower than it. So, $4!=4\times3\times2\times1$ $$\require{cancel}\eqalign{\dfrac{n!}{r!(n-r)!}&=\text{Number of combinations}\\\dfrac{10!}{4!(10-4)!}&=\\\dfrac{10!}{4!\times6!}&=\dfrac{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{(4\times3\times2\times1)\times(6\times5\times4\times3\times2\times1)} \\\text{Reduce fractions before }&\text{ doing the multiplication!}\\\dfrac{10\times9\times8\times7\times\cancel{6\times5\times4\times3\times2\times1}}{(4\times3\times2\times1) \times (\cancel{6\times5\times4\times3\times2\times1)}}&=\dfrac{2880}{24}=120 \text{ Possible combinations}}$$ | Find the button that looks like $\boxed{_nC_r}$ on your calculator. It might be a "second" function that is written above one of your keys. If you have a graphing calculator, it might be in the Math/Prob menu. On most calculators, you will enter n, then press the button, then enter r. In this case, you will enter: $\boxed{10}\quad \boxed{_nC_r}\quad\boxed{4}\quad\boxed{\text{Enter}}$ Your calculator should yield: 120
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Can you believe that you can make 120 possible quartets? (By the way, if you really did want to know all of the permutations (how many combinations, including the ones with the same people but playing different instruments), there are 5,040 possible quartet permutations!)
Once you get used to working with factorials, the trickiest part of finding combinations is figuring out if you want combinations or permutations. Once you get that figured out, you figure out your set of options, the number of options you want and plug those values into the formula (or the calculator) and you'll get your number of combinations!
Practice Problems:
Combinations (Making sets, of a certain size, from another set)
Calculate the number of possible combinations:
- You have a set of 10 objects. How many different sets of 3 objects can you make (order does not matter)?
- You have a set of 6 objects. How many different sets of 5 objects can you make (order does not matter)?
- You have a set of 4 objects. How many different sets of 4 objects can you make (order does not matter)?
- You are shopping for t-shirts. The store has 5 different colors. You are buying 3 different colored shirts. How many different combinations of shirts can you choose?
- You making a sandwich. You can choose 5 of 12 different toppings (meats, veggies, etc). How many different sandwiches can you make?
- In an ice cream shop, there are 20 possible ice cream flavors. You are getting a double scoop cone (you must choose two different flavors). How many different cones can you get?
- In the same ice cream shop, how many different flavered three-scoop cones can you get?
- In that same ice cream shop, if you can also choose from 2 different types of cones (sugar or cake), how many possible three-scoop cones can you get?