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# Combination Problems

## Using n!/r!(n - r)! to calculate permutations

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Practice Combination Problems
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Evaluate Combinations Using Combination Notation

Do you know how to use combination notation? Take a look at this dilemma.

Evaluate the following combination.

Find 8C3\begin{align*}{{_8}C{_3}}\end{align*}

To figure this out, you will need to understand combination notation. Pay attention and you will know how to evaluate this combination by 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.

When evaluating a combination, you can use a tree diagram. Use a tree diagram can be time consuming, combination notation is a much simpler option.

To use combination notation, you must first understand factorials. Do you remember factorials?

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

Take a look at this one.

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*}

The answer is 120.

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.

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*}

Here is another one.

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.

Evaluate each combination.

#### Example A

Find 6C3\begin{align*}{{_6}C{_3}}\end{align*}

Solution: 20\begin{align*}20\end{align*} arrangements

#### Example B

Find 9C2\begin{align*}{{_9}C{_2}}\end{align*}

Solution: 36\begin{align*}36\end{align*} arrangements

#### Example C

Find 5C4\begin{align*}{{_5}C{_4}}\end{align*}

Solution: 5\begin{align*}5\end{align*} arrangements

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

Find 8C3\begin{align*}{{_8}C{_3}}\end{align*}

First, we can write out the numerator.

8×7×6×5×4×3×2×1(3×2×1)(5×4×3×2×1)\begin{align*}\frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(5 \times 4 \times 3 \times 2 \times 1)}\end{align*}

Next, we simplify.

8×7×63×2×1\begin{align*}\frac{8 \times 7 \times 6}{3 \times 2 \times 1}\end{align*}

56\begin{align*}56\end{align*} arrangements

This is our answer.

### Vocabulary

Combination
an arrangement of items or events where the order is not 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.

Write the following situation using combination notation. Then evaluate it.

Sixteen students went to the park. Four students could ride in four cars. How many different combinations of students could there be?

Solution

First, use combination notation.

Find 16C4\begin{align*}{{_16}C{_4}}\end{align*}

Now we can evaluate the combination by simplifying first.

16×15×14×134×3×2×1\begin{align*}\frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1}\end{align*}

43,68024\begin{align*}\frac{43,680}{24}\end{align*}

There can be 1,820\begin{align*}1,820\end{align*} different arrangements.

### Practice

Directions: Evaluate each combination.

1. Find 5C2\begin{align*}{{_5}C{_2}}\end{align*}
2. Find 6C5\begin{align*}{{_6}C{_5}}\end{align*}
3. Find 7C2\begin{align*}{{_7}C{_2}}\end{align*}
4. Find 7C3\begin{align*}{{_7}C{_3}}\end{align*}
5. Find 8C2\begin{align*}{{_8}C{_2}}\end{align*}
6. Find 6C4\begin{align*}{{_6}C{_4}}\end{align*}
7. Find 9C2\begin{align*}{{_9}C{_2}}\end{align*}
8. Find 9C4\begin{align*}{{_9}C{_4}}\end{align*}
9. Find 8C3\begin{align*}{{_8}C{_3}}\end{align*}
10. Find 4C4\begin{align*}{{_4}C{_4}}\end{align*}

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?

### Vocabulary Language: English

combination

combination

Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.
combination notation

combination notation

Combination notation has the forms nCr and c(n, r) where n is the number of different units to choose from and r is the number of units in each group.
combinatorics

combinatorics

Combinatorics is the study of permutations and combinations.
factorial

factorial

The factorial of a whole number $n$ is the product of the positive integers from 1 to $n$. The symbol "!" denotes factorial. $n! = 1 \cdot 2 \cdot 3 \cdot 4...\cdot (n-1) \cdot n$.
n value

n value

When calculating permutations with the TI calculator, the n value is the number of objects from which you are choosing.
permutation notation

permutation notation

Permutation notation is the form  nPr  or P(n, r), and indicates the number of ways that n objects can be ordered into groups of r items each.
TI-84

TI-84

The TI-84 calculator is a graphing calculator produced by Texas Instruments and is considered an “industry standard” for more advanced calculations.