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# 11.3: Combinations

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Big Race

“Come on girls, you can help me to make this list,” Ms. Kelley called to Carey and Telly.

It was the girls third day of work and they were beginning to help Ms. Kelley to organize a big bike race. The shop was hosting a group of bikers from all around the area to help raise money for charity. Bikers from all over had already registered and now it was just a matter of Ms. Kelley organizing the list into a starting order.

“How do you decide who gets to race in the front?” Carey asked.

“I guess I do,” Carey said.

“Well, it is totally random. This isn’t a professional race, so it doesn’t matter who starts where. Everyone will get an equal chance and besides, all of the money raised goes to charity,” Ms. Kelly explained.

Carey looked at Telly.

“A combination, not a permutation,” Carey said.

Telly got it this time. After the situation with the locks she now understood the definition of a permutation.

Do you understand how a permutation is different from a combination? Why did Carey say that this was a combination?

What You Will Learn

In this lesson, you will learn how to do the following skills.

• Recognize combinations as arrangements in which order is not important.
• Count all combinations of n\begin{align*}n\end{align*} objects or events.
• Count combinations of n\begin{align*}n\end{align*} objects taken r\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

Order is important for some groups of items but not important for others. For example, consider a list of the words: POTS, STOP, SPOT, and TOPS.

• For the spelling of each individual word, order is important. The words POTS, STOP, SPOT, and TOPS 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 POTS, STOP, SPOT, TOPS, or another order, such as STOP, SPOT, TOPS, POTS, or a third order, such as TOPS, POTS, SPOT, STOP – makes no difference. As long as the list includes all 4 words, the order of the 4 words doesn’t matter.

A combination is an arrangement 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.

Think about a pizza. It doesn’t matter which order you put on the toppings once they are all on there. You can put a combination of toppings on a pizza. Let’s look at the example.

Example

Six people – Larry, Sherry, Terri, Carrie, Mary, and Harry all want to ride in a rowboat that can hold only 4 passengers. How many different groups of 4 passengers can ride in the boat?

Step 1: Write out a single order.

Larry, Sherry, Terri, Harry

Step 2: Now rearrange the order. Did changing the order of the items change the outcome? If so, then order matters.

Sherry, Harry, Larry, Terri \begin{align*}\Longleftarrow\end{align*} different order, same 4 passengers

Order does NOT matter for this problem. Use combinations.

Example

Four tap dancers are entered in the Star Power Talent Show – Debbie, Maurice, Minnie, and Ronnie. The 4 will appear separately on stage. In how many different ways can the 4 be scheduled to appear on stage?

Step 1: Write out a single order.

Debbie, Ronnie, Maurice, Minnie

Step 2: Now rearrange the order. Did you change the outcome? If so, then order matters.

Maurice, Debbie, Minnie, Ronnie \begin{align*}\Longleftarrow\end{align*} different order, meaning is DIFFERENT

Order DOES matter for this problem. Use permutations, not combinations.

II. Count All Combinations of n\begin{align*}\underline{n}\end{align*} Objects or Events

When solving problems, we can use combinations to solve problems when order is not important. One way to find the number of combinations is to use a tree diagram.

Example

For his top tennis doubles team, Coach Yin is considering 3 players: Joyce, Rose, and Nica. How many different doubles teams can the coach consider?

The first tree diagram shows all 6 permutations of the 3 players. But order doesn’t matter in this problem. For example, the team of Joyce-Rose is no different than the team of Rose-Joyce.

So in the second tree diagram we cross out all outcomes that are repeats. This leaves 3 combinations that are not repeats.

Joyce-Rose, Joyce-Nica, Rose-Nica

This method of making a tree diagram and crossing out repeats is reliable, but it is not the only way to find combinations. You can list them out.

Example

How many different violin duos can Ben, Jen, Ren, Wen, and Ken form?

Combination{\color{red}Ben,} {\color{red}Jen}, Ren, Wen, Ken{\color{red}Ben}, Jen, {\color{red} Ren}, Wen, Ken{\color{red}Ben}, Jen, Ren, {\color{red}Wen}, Ken{\color{red}Ben}, Jen, Ren, Wen, {\color{red}Ken}List{\color{red}Ben-Jen}{\color{red}Ben-Ren}{\color{red}Ben-Wen}{\color{red}Ben-Ken}\begin{align*} & \underline{\text{Combination}} && \underline{\text{List}}\\ & \text{{\color{red}Ben,} {\color{red}Jen}, Ren, Wen, Ken} && \text{{\color{red}Ben-Jen}}\\ & \text{{\color{red}Ben}, Jen, {\color{red} Ren}, Wen, Ken} && \text{{\color{red}Ben-Ren}}\\ & \text{{\color{red}Ben}, Jen, Ren, {\color{red}Wen}, Ken} && \text{{\color{red}Ben-Wen}}\\ & \text{{\color{red}Ben}, Jen, Ren, Wen, {\color{red}Ken}} && \text{{\color{red}Ben-Ken}}\\\end{align*}

Step 2: You’ve covered all combinations that begin with Ben. Now go through all combinations that begin with Jen, Ren, and Wen.

CombinationBen, Jen, Ren, Wen, KenBen, Jen, Ren, Wen, KenBen, Jen, Ren, Wen, KenBen, Jen, Ren, Wen, KenBen, {\color{red}Jen}, {\color{red}Ren}, Wen, KenBen, {\color{red}Jen}, Ren, {\color{red} Wen}, KenBen, {\color{red}Jen}, Ren, Wen, {\color{red}Ken}Ben, Jen, {\color{red}Ren}, {\color{red}Wen}, KenBen, Jen, {\color{red}Ren}, Wen, {\color{red}Ken}Ben, Jen, Ren, {\color{red}Wen}, {\color{red}Ken}ListBen-JenBen-RenBen-WenBen-Ken{\color{red}Jen-Ren}{\color{red}Jen-Wen}{\color{red}Jen-Ken}{\color{red}Ren-Wen}{\color{red}Ren-Ken}{\color{red}Wen-Ken}\begin{align*} & \underline{\text{Combination}} && \underline{\text{List}}\\ & \text{Ben, Jen, Ren, Wen, Ken} && \text{Ben-Jen}\\ & \text{Ben, Jen, Ren, Wen, Ken} && \text{Ben-Ren}\\ & \text{Ben, Jen, Ren, Wen, Ken} && \text{Ben-Wen}\\ & \text{Ben, Jen, Ren, Wen, Ken} && \text{Ben-Ken}\\ & \text{Ben, {\color{red}Jen}, {\color{red}Ren}, Wen, Ken} && \text{{\color{red}Jen-Ren}}\\ & \text{Ben, {\color{red}Jen}, Ren, {\color{red} Wen}, Ken} && \text{{\color{red}Jen-Wen}}\\ & \text{Ben, {\color{red}Jen}, Ren, Wen, {\color{red}Ken}} && \text{{\color{red}Jen-Ken}}\\ & \text{Ben, Jen, {\color{red}Ren}, {\color{red}Wen}, Ken} && \text{{\color{red}Ren-Wen}}\\ & \text{Ben, Jen, {\color{red}Ren}, Wen, {\color{red}Ken}} && \text{{\color{red}Ren-Ken}}\\ & \text{Ben, Jen, Ren, {\color{red}Wen}, {\color{red}Ken}} && \text{{\color{red}Wen-Ken}}\\ \end{align*}

Your list is now complete. In all, there are 10 combinations.

III. Count Combinations of n\begin{align*}\underline{n}\end{align*} Objects Taken r\begin{align*}\underline{r}\end{align*} at a Time

In the last section you learned about permutations. Remember that a permutation is an arrangement of objects or events that are organized in a particular way. We can agree that a combination is an arrangement of objects or events where order does not matter. When we worked with permutations, we used factorials to show how a number of arrangements were taken so many at a time. Let’s review factorials.

A factorial is a special number that represents the product of a set of values in descending order.

Example

5!

To evaluate 5! We can say that this is the product of values starting with 5 in descending order.

5×4×3×2×1=120\begin{align*}5 \times 4 \times 3 \times 2 \times 1 = 120\end{align*}

We can use factorials and combination notation to evaluate combinations without using lists or tree diagrams. Let’s take a look at how this works.

IV. Evaluate Combinations Using Combination Notation

Just like you learned how to use notation and factorials to figure out permutations, you can learn a similar calculation method for combinations.

The notation for combinations is similar to the notation for permutations. To represent the number of combinations there are for 6 items taken 4 at a time, write:

6C4 \begin{align*}{\color{red}_6}C{\color{blue}_4} \ \Longleftarrow \end{align*} 6 items taken 4 at a time

In general, combinations are written as:

nCr n\begin{align*}{\color{red}_n}C{\color{blue}_r} \ \Longleftarrow \color{red}n\end{align*} items taken r\begin{align*}\color{blue}r\end{align*} at a time

To compute nCr\begin{align*}{{_n}C{_r}}\end{align*} use the formula:

nCr=n!r!(nr)!\begin{align*}{\color{red}_n}C{\color{blue}_r}=\frac{{\color{red}n}!}{{\color{blue}r!}({\color{red}n}-{\color{blue}r})!}\end{align*}

Example

Find 5C2\begin{align*}{{_5}C{_2}}\end{align*}

Step 1: Understand what 5C2\begin{align*}{_5}C{_2}\end{align*} means.

5C2 \begin{align*}{\color{red}_5}C{\color{blue}_2} \ \Longleftarrow \end{align*} 5 items taken 2 at a time

Step 2: Set up the problem.

5C2=5!2!(52)!\begin{align*}{\color{red}_5}C{\color{blue}_2}=\frac{{\color{red}5}!}{{\color{blue}2!}({\color{red}5}-{\color{blue}2})!}\end{align*}

Step 3: Fill in the numbers and simplify.

5C2=5!2!(3!)=5×42×3×2×12×1×(3×2×1)=5×21=10\begin{align*}{{_5}C{_2}}=\frac{5!}{2! (3!)}=\frac{5 \times \overset{2}{\cancel{4} } \times \cancel{3 \times 2 \times 1}}{\cancel{2} \times \cancel{1} \times \cancel{(3 \times 2 \times 1)}}=\frac{5 \times 2}{1}=10\end{align*}

There are 10 different possible combinations.

Now that you understand this notation, let’s go back to the problem from the introduction.

## Real-Life Example Completed

The Big Race

Here is the original problem once again. Reread it and then explain why Carey and Telly agree that this is a combination not a permutation.

“Come on girls, you can help me to make this list,” Ms. Kelley called to Carey and Telly.

It was the girls third day of work and they were beginning to help Ms. Kelley to organize a big bike race. The shop was hosting a group of bikers from all around the area to help raise money for charity. Bikers from all over had already registered and now it was just a matter of Ms. Kelley organizing the list into a starting order.

“How do you decide who gets to race in the front?” Carey asked.

“I guess I do,” Carey said.

“Well, it is totally random. This isn’t a professional race, so it doesn’t matter who starts where. Everyone will get an equal chance and besides, all of the money raised goes to charity,” Ms. Kelly explained.

Carey looked at Telly.

“A combination, not a permutation,” Carey said.

Telly got it this time. After the situation with the locks she now understood the definition of a permutation.

Be sure to explain your thinking.

Solution to Real – Life Example

The key here is that the order of the bikers does not matter. Ms. Kelley said that it is random. Anyone can be in the front and therefore this is a combination. A combination is a series where order does not matter. If this was a situation where order did matter, then it would be considered a permutation.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Combination
an arrangement of items or events where the order is not important.
Permutation
an arrangement of items or events where the order is important.
Factorial
a special number which represents the product of numbers in descending order.

## Time to Practice

Directions: Write whether order is important for each problem. Write whether you would be likely to use combinations or permutations to solve the problem.

1. At Dudley’s Dude Ranch there are 6 riders but only 4 horses. How many different ways can a group of 4 go out on ride?
2. With 4 laps to go, Dale Earnhardt Jr., Robbie Gordon, Kyle Busch, and Kasey Kahne are all in contention to win a NASCAR race. In how many different ways can the drivers finish the race? Do you need to find permutations or combinations to solve this problem?
3. Ace, King, Queen, Jack, Ten, and Nine of Clubs are face down on a table. How many different 3-card hands can you draw all at once?
4. 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?
5. How many different 4-horn bands can you choose from a class of 10 horn players?
6. Eight candidates are running in the primary elections for president. How many president and vice president pairs are possible?
7. Fifteen students compete in the Geography Bee. How many different winners, 2nd place, and 3rd place finishers can there be?
8. Nine people want to ride on the banana boat but there are only 4 life jackets. How many different groups can ride on the banana boat at one time?
9. The 5 last people at a movie must compete for the last 3 empty seats. How many different groups of 3 can sit and watch the movie?

Directions: Use what you have learned to figure out combinations.

1. Leah collected 3 different flowers for a bouquet – a rose, a tulip, and a daffodil. How many 2-flower bouquets can she make? List the bouquets.
2. Leah added a lily to her flowers. How many 2-flower bouquets can she make out of a rose, a tulip, a daffodil, and a lily? List the bouquets.
3. How many 3-flower bouquets can Leah make out of a rose, a tulip, a daffodil, and a lily? List the bouquets.
4. How many 2-flower bouquets can Leah make out of a rose, a tulip, a daffodil, a lily, and a violet?
5. How many 3-flower bouquets can Leah make out of a rose, a tulip, a Daffodil, a lily, and a violet?
6. At Dudley’s Dude Ranch there are 5 dudes who want to ride – Peg, Greg, Meg, Sue, and Drew – but only 4 horses. How many different 4-horse groups can go out for a ride?

Directions: Use the formula to figure out the different combinations.

1. How many different color pairs are there among red, orange, yellow, green, and blue?
2. How many different sets of 3 colors are there among red, orange, yellow, green, and blue?
3. How many different color pairs are there among red, orange, yellow, green, blue, and purple?
4. How many different sets of 3 colors are there among red, orange, yellow, green, blue, and purple?
5. How many different sets of 3 colors are there among red, orange, yellow, green, blue, purple, and white?
6. Ten tennis players are on the Davis Cup Team. Only two players can play in the doubles finals. How many different doubles teams could play in the finals?

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