12.7: Combinations
Introduction
Decorating the Stage
The decorating committee is getting the stage ready for the Talent Show. There was a bunch of different decorating supplies ordered, and the students on the committee are working on figuring out the best way to decorate the stage.
They have four different colors of streamers to use to decorate.
Red
Blue
Green
Yellow
“I think four is too many colors. How about if we choose three of the four colors to decorate with?” Keith asks the group.
“I like that idea,” Sara chimes in. “How many ways can we decorate the stage if we do that?”
The group begins to figure this out on a piece of paper.
Combinations are arrangements where order does not make a difference. The decorating committee is selecting three colors from the possible four options. Therefore, the order of the colors doesn’t matter.
Combinations are the way to solve this problem. Look at the information in this lesson to learn how to figure out the possible combinations.
What You Will Learn
In this lesson you will learn how to:
- Recognize combinations as arrangements in which order is not important.
- Count all combinations of \begin{align*}n\end{align*} objects or events
- Count combinations of \begin{align*}n\end{align*} objects taken \begin{align*}r\end{align*} at a time
- Evaluate combinations using combination notation.
Teaching Time
I. Recognize Combinations as Arrangements in Which Order is Not Important
In the last section, you saw that order is important for some groups of items but not important for others. For example, consider a list of three words: HOPS, SHOP, and POSH.
- For the spelling of each individual word, order is important. The words HOPS, SHOP, and POSH all use the same letters, but spell out very different words.
- For the list itself, order is not important. Whether the words are presented in one order–such as HOPS, SHOP, POSH, or another order, such as SHOP, POSH, HOPS, or a third order, such as POSH, HOPS, SHOP–makes no difference. As long as the list includes all 3 words, the order of the 3 words doesn’t matter.
A combination is a collection of items in which order, or how the items are arranged, is not important. The collection of one order of the items is not functionally different than any other order.
Combinations and permutations are related. To solve problems in which order matters, you use permutations. To solve problems in which order does NOT matter, use combinations.
Let’s look at an example.
Example
The winning 3-digit lottery numbers are drawn from a drum as 641, 224, and 806. Does order matter in the way the three winning numbers are drawn?
Step 1: Write out a single order.
641, 224, 806
Step 2: Now rearrange the order. Did you change the outcome? If so, then order matters.
\begin{align*}224, 806, 641\Longleftarrow \end{align*} different order, same 3 winning numbers
Order does NOT matter for this problem. Use combinations.
Write the difference between combinations and permutations down in your notebook.
Example
A bag has 4 marbles: red, blue, yellow, and green. In how many different ways can you reach into the bag and draw out 1 marble, then return the marble to the bag and draw out a second marble?
Step 1: Write out a single order.
red, blue
Step 2: Now rearrange the order. Did you change the outcome? If so, then order matters.
blue, red \begin{align*}\Longleftarrow \end{align*} different order, meaning is DIFFERENT
Order DOES matter for this problem. Use permutations.
12K. Lesson Exercises
Write whether you would use combinations or permutations for each example.
- Cesar the dog-walker has 5 dogs but only 3 leashes. How many different ways can Cesar take a walk with groups of 3 dogs at once?
- Five different horses entered the Kentucky Derby. In how many different ways can the horses finish the race?
- How many different 5-player teams can you choose from a total of 8 basketball players?
Take a few minutes to check your answers with a friend.
II. Count All Combinations of \begin{align*}n\end{align*} Objects or Events
Once you figure out if you are going to be using permutations or combinations, it is necessary to count the combinations.
There are several different ways to count combinations. When counting, try to keep the following in mind:
- Go one by one through the items. Don’t stop your list until you’ve covered every possible link of one item to all other items.
- Keep in mind that order doesn’t matter. For combinations, there no difference between \begin{align*}AB\end{align*} and \begin{align*}BA\end{align*}. So if both \begin{align*}AB\end{align*} and \begin{align*}BA\end{align*} are on your list, cross one of the choices off your list.
- Check your list for repeats. If you accidentally listed a combination more than once, cross the extra listings off your list.
Example
James needs to choose a 2-color combination for his intramural team t-shirts. How many different 2-color combinations can James make out of red, blue, and yellow?
One way to find the number of combinations is to make a tree diagram. Here, if red is chosen as one color, that leaves only blue and yellow for the second color.
The diagram shows all 6 permutations of the 3 colors. But wait–since we are counting COMBINATIONS here order doesn’t matter.
So in this tree diagram we will cross out all outcomes that are repeats. For example, the first red-blue is no different from blue-red, so we’ll cross out blue-red.
In all, there are 3 combinations that are not repeats.
This method of making a tree diagram and crossing out repeats is reliable, but it is not the only way to find combinations.
Let’s look at another example.
Example
James has added a fourth color, green, to choose from in selecting a 2-color combination for his intramural team. How many different 2-color combinations can James make out of red, blue, yellow, and green?
Step 1: Write the choices. Match the first choice, red, with the second, blue. Add the combination, red-blue, to your list. Match the other choices in turn. Add the combinations to your list.
Step 2: Now move to the second choice, blue. Match blue up with every possible partner other than red, since we already included all of the combinations involving red. Add the combinations to your list.
Step 3: Now move to the third choice, yellow. There is only one new combination left to match it with. Add the combination to your list.
Your list is now complete. There are 6 combinations.
II. Count All Combinations of \begin{align*}n\end{align*} Objects Taken \begin{align*}r\end{align*} at a Time
Sometimes, you won’t want to use all of the possible options in the combination. Think about it as if you have 16 flavors of ice cream, but you only want to use three flavors at a time. This is an example where there are 16 flavors to work with, but you can only use three at a time. With an example like this one, you are looking for combinations of object where only a certain number of them are used in any one combination.
This happens a lot with teams. Let’s look at an example.
Example
How many different 2-player soccer teams can Jean, Dean, Francine, Lurleen, and Doreen form?
Step 1: Start with Jean. Add all combinations that begin with Jean to your list.
\begin{align*}&\underline{\text{Combination}} \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \underline{\text{List}}\\ &\text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Dean}\\ &\text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Francine}\\ &\text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Lurleen}\\ &\text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Doreen}\end{align*}
Step 2: You’ve covered all combinations that begin with Jean. Now go through all combinations that begin with Dean, Francine, and Lurleen.
\begin{align*}&\underline{\text{Combination}} \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \underline{\text{List}}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Dean}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Francine}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Lurleen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Doreen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Dean-Francine}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Dean-Lurleen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Dean-Doreen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Francine-Lurleen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Francine-Doreen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Lurleen-Doreen}\end{align*}
Your list is now complete. There are 10 combinations.
Example
How many different 3-player soccer teams can Jean, Dean, Francine, Lurleen, and Doreen form?
Use the process above to go through all of the combinations.
\begin{align*}&\underline{\text{Combination}} \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \underline{\text{List}}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Dean-Francine}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Dean-Lurleen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Dean-Doreen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Francine-Lurleen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Francine-Doreen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Jean-Lurleen-Doreen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Dean, Francine-Lurleen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Dean-Francine-Doreen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Dean-Lurleen-Doreen}\\ & \text{Jean, Dean, Francine, Lurleen, Doreen} \qquad \text{Francine-Lurleen-Doreen}\end{align*}
Your list is now complete. There are 10 combinations.
Try a few of these on your own.
12L. Lesson Exercises
- On Monday Cesar the dog-walker has 3 dogs–Looie, Huey, and Dewey-but only 2 leashes. How many different ways can Cesar take a walk with 2 dogs? List the ways.
- On Tuesday Cesar has 4 dogs-Looie, Huey, Dewey, and Stewie–but only 2 leashes. How many different ways can Cesar take a walk with 2 dogs? List the ways.
- On Wednesday Cesar has 4 dogs-Looie, Huey, Dewey, and Stewie–but now has 3 leashes. How many different ways can Cesar take a walk with 3 dogs? List the ways.
Take a few minutes to discuss your findings with a partner. Share your method of finding all of the possible combinations.
IV. Evaluate Combinations Using Combination Notation
We can use a formula to help us to calculate combinations. This is very similar to the work that you did in the last section with factorials and permutations.
Example
Suppose you have 5 marbles in a bag–red, blue, yellow, green, and white. You want to know how many combinations there are if you take 3 marbles out of the bag all at the same time. In combination notation you write this as:
\begin{align*}{_5}C_3 \Longleftarrow 5 \ \text{items taken 3 at a time}\end{align*}
In general, combinations are written as:
\begin{align*}{_n}C_r \Longleftarrow n \ \text{items taken} \ r \ \text{at a time}\end{align*}
To compute \begin{align*}{_n}C_r\end{align*} use the formula:
\begin{align*}{_n}C_r = \frac{n!}{r!(n - r)!}\end{align*} This may seem a bit confusing, but it isn’t. Notice that the factorial symbol is used with the number of object \begin{align*}(n)\end{align*} and the number taken at any one time \begin{align*}(r)\end{align*}. This helps us to understand which value goes where in the formula.
Now let’s look at applying the formula to the example.
For \begin{align*}{_5}C_3\end{align*}:
\begin{align*}{_5}C_3 = \frac{5!}{3!(5 - 3)!} = \frac{5!}{3! 2!}\end{align*} Simplify.
\begin{align*}{_5}C_3 = \frac{5 (4)(3)(2)(1)}{(3 \cdot 2 \cdot 1)(2 \cdot 1)} = \frac{120}{12} = 10\end{align*}
There are 10 possible combinations.
Example
Find \begin{align*}{_6}C_2\end{align*}
Step 1: Understand what \begin{align*}{_6}C_2\end{align*} means.
\begin{align*}{_6}C_2 \Longleftarrow 6 \ \text{items taken 2 at a time}\end{align*}
Step 2: Set up the problem.
\begin{align*}{_6}C_2 = \frac{6!}{2!(6 -2)!}\end{align*} Step 3: Fill in the numbers and simplify.
\begin{align*}{_6}C_2 = \frac{6(5)(4)(3)(2)(1)}{(2 \cdot 1)(4 \cdot 3 \cdot 2 \cdot 1)} = \frac{720}{48} = 15\end{align*}
There are 15 possible combinations.
12M. Lesson Exercises
Find the number of combinations in each example.
- \begin{align*}{_5}C_2\end{align*}
- \begin{align*}{_4}C_3\end{align*}
- \begin{align*}{_6}C_4\end{align*}
Check your answers with a friend.
Copy down the formula for figuring out combinations in your notebook
Real–Life Example Completed
Decorating the Stage
Here is the original problem once again. Reread it and then figure out the decorations.
The decorating committee is getting the stage ready for the Talent Show. There was a bunch of different decorating supplies ordered, and the students on the committee are working on figuring out the best way to decorate the stage.
They have four different colors of streamers to use to decorate.
Red
Blue
Green
Yellow
“I think four is too many colors. How about if we choose three of the four colors to decorate with?” Keith asks the group.
“I like that idea,” Sara chimes in. “How many ways can we decorate the stage if we do that?”
The group begins to figure this out on a piece of paper.
Combinations are arrangements where order does not make a difference. The decorating committee is selecting three colors from the possible four options. Therefore, the order of the colors doesn’t matter.
We can use combination notation to figure out this problem.
\begin{align*}{_4}C_3 = \frac{4!}{3!(4 - 3)!} = \frac{4(3)(2)(1)}{(3 \cdot 2 \cdot 1)(1)}= \frac{24}{6} = 4\end{align*}
There are four possible ways to decorate the stage.
Now that the students have this information, they can look at their color choices and vote on which combination they like best.
Vocabulary
Here are the vocabulary words used in this lesson.
- Combination
- an arrangement of objects or events where order does not matter.
- Permutations
- an arrangement of objects or events where the order does matter.
Technology Integration
Time to Practice
Directions: Write whether you are more likely to use permutations or combinations for each of the following examples.
1. A bag has 4 marbles: red, blue, yellow, and green. In how many different ways can you reach into the bag and draw out 2 marbles at once and drop them in a cup?
2. A bag contains 5 slips of paper with letters \begin{align*}A, B, C, D\end{align*}, and \begin{align*}E\end{align*} written on them. Pull out one slip, mark down the letter and replace it in the bag. Do this 3 times so you have written 3 letters. How many different ways can you write the 3 letters?
3. Eight candidates are running for the 4-person Student Council. How many different Student Councils are possible?
4. Mario’s gym locker uses the numbers 14, 6, and 32. How many different arrangements of the three numbers must Mario try to be sure he opens his locker?
5. Five horn players are running for 2 seats in a jazz band. How many different ways can the two horn players be chosen?
Directions: Use what you have learned about combinations to answer each question.
6. The Ace, King, Queen, and Jack of Spades are face down on a table. Draw three cards all at once. How many different 3-card hands can you draw?
7. How many different 4-player teams can you choose from a total of 5 volleyball players:
Andy, Randi, Sandy, Mandy, and Chuck?
8. How many different 3-player teams can you choose from a total of 5 volleyball players:
Andy, Randi, Sandy, Mandy, and Chuck?
9. A bag contains 6 slips of paper with letters \begin{align*}A, B, C, D, E\end{align*}, and \begin{align*}F\end{align*} written on them. Pull out 4 slips. How many different 4-slip combinations can you get?
Directions: Evaluate each factorial.
10. 5!
11. 4!
12. 3!
13. 8!
14. 9!
15. 6!
Directions: Evaluate each combination using combination notation.
16. \begin{align*}{_7} C_2\end{align*}
17. \begin{align*}{_7} C_6\end{align*} 18. \begin{align*}{_8} C_4\end{align*}
19. \begin{align*}{_9} C_6\end{align*}
20. \begin{align*}{_8} C_3\end{align*}
21. \begin{align*}{_{10}}C_7 \end{align*}
22. \begin{align*}{_{12}}^*C_9\end{align*} 23. \begin{align*}{_{11}}^*C_9\end{align*}
24. \begin{align*}{_{16}}^*C_{14}\end{align*}
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