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# Combinations

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Progress
Practice Combinations
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Recognizing Combinations

Have you ever been in a bike race? Take a look at this dilemma.

“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? Pay attention and you will be able to answer these questions at the end of the Concept.

### Guidance

Order is important for some groups of items but not important for others. 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.

Sometimes order does matter. When the order does matter, then use a permutation.

Take a look at this situation.

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 $\Longleftarrow$ different order, same 4 passengers

Order does NOT matter for this problem. Use combinations.

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 $\Longleftarrow$ different order, meaning is DIFFERENT

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

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.

Take a look at this situation.

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.

Evaluate each combination.

#### Example A

How many different letter arrangements can you make if you have six letters but only use three at a time?

Solution: $20$ arrangements

#### Example B

How many different letter arrangements can you make if you have six letters, but only use four at a time?

Solution: $15$ arrangements

#### Example C

How many different letter arrangements can you make if you have seven letters, but only use three at a time?

Solution: $35$ arrangements

Now let's go back to the dilemma from the beginning of the Concept.

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

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.

### Guided Practice

Here is one for you to try on your own.

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

Solution

$& \underline{\text{Combination}} && \underline{\text{List}}\\& \text{{\color{red}Ben,} {\color{red}Jen}, Ren, Wen, Ken} && \text{{\color{red}Ben-Jen}}\\& \text{{\color{red}Ben}, Jen, {\color{red} Ren}, Wen, Ken} && \text{{\color{red}Ben-Ren}}\\& \text{{\color{red}Ben}, Jen, Ren, {\color{red}Wen}, Ken} && \text{{\color{red}Ben-Wen}}\\& \text{{\color{red}Ben}, Jen, Ren, Wen, {\color{red}Ken}} && \text{{\color{red}Ben-Ken}}\\$

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

$& \underline{\text{Combination}} && \underline{\text{List}}\\& \text{Ben, Jen, Ren, Wen, Ken} && \text{Ben-Jen}\\& \text{Ben, Jen, Ren, Wen, Ken} && \text{Ben-Ren}\\& \text{Ben, Jen, Ren, Wen, Ken} && \text{Ben-Wen}\\& \text{Ben, Jen, Ren, Wen, Ken} && \text{Ben-Ken}\\& \text{Ben, {\color{red}Jen}, {\color{red}Ren}, Wen, Ken} && \text{{\color{red}Jen-Ren}}\\& \text{Ben, {\color{red}Jen}, Ren, {\color{red} Wen}, Ken} && \text{{\color{red}Jen-Wen}}\\& \text{Ben, {\color{red}Jen}, Ren, Wen, {\color{red}Ken}} && \text{{\color{red}Jen-Ken}}\\& \text{Ben, Jen, {\color{red}Ren}, {\color{red}Wen}, Ken} && \text{{\color{red}Ren-Wen}}\\& \text{Ben, Jen, {\color{red}Ren}, Wen, {\color{red}Ken}} && \text{{\color{red}Ren-Ken}}\\& \text{Ben, Jen, Ren, {\color{red}Wen}, {\color{red}Ken}} && \text{{\color{red}Wen-Ken}}\\$

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

### Practice

Directions: Solve each combination.

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 top three drivers finish?
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 ways can three people be chosen as winners?
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?
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?
3. How many 3-flower bouquets can Leah make out of a rose, a tulip, a daffodil, and a lily?
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?