Suppose there are fifteen students in a classroom, and you are looking to build a team of five students for a tug-of-war. How many different groups of five students could you create?

After the following lesson, you should have no problem solving combination question of this type.

### Calculating Combinations

In other lessons, we have spent quite a bit of time learning how to count **permutations**, which are arrangements of items that take order into account. Now we will be considering **combinations**, which are groups of items only concerned with which items are in the arrangement, regardless of the order.

Naturally there are fewer unique combinations of a particular group of items than there are permutations, since there may be many permutations of the same single combination of items. In fact, the formula for counting combinations without repetition is actually just a modification of the formula for permutations where the denominator is increased. Take a look at the two formulas side-by-side:

\begin{align*}Permutations: \ \frac{n!}{(n-r)!} \qquad Combinations: \ \frac{n!}{(n-r)! \times r!} \end{align*}

Where \begin{align*}n =\end{align*}

Notice how the combination formula is the same, with only the addition of the \begin{align*}r!\end{align*}

In summary, to calculate the number of **combinations** possible from \begin{align*}n\end{align*}

#### Real-World Application: Ice Cream

The local ice cream shop carries twelve flavors. How many different three-scoop bowls are possible?

There are twelve flavors, so \begin{align*}n=12\end{align*}

\begin{align*}& \qquad \qquad \qquad \qquad \ \ \frac{12!}{(12-3)! \times 3!}\\
& \qquad \qquad \qquad \qquad \qquad \frac{12!}{9! \times 3!}\\
& \frac{12 \times 11 \times 10 \times {\color{red}\bcancel{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}}{{\color{red}\bcancel{(9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}}(3 \times 2 \times 1)} \ (\text{cancel terms})\\
& \qquad \qquad \qquad \qquad \ \ \qquad \frac{1320}{6}\\
& \qquad \qquad \qquad \qquad 220 \ different \ flavors\end{align*}

#### Real-World Application: Forming Groups

If there are thirteen students in a room, how many ways can a group of five be formed from them?

There are thirteen students to choose from, so \begin{align*}n = 13\end{align*}

We are choosing five at a time, so \begin{align*}r = 5\end{align*}

\begin{align*}& \qquad \quad \frac{13!}{(13-5)! \times 5!}\\
& \qquad \qquad \frac{13!}{(8!)(5!)}\\
& \quad \ \frac{13 \times 12 \times 11 \times 10 \times 9}{5!}\\
& \qquad \qquad \frac{154440}{120}\\
& 1287 \ \text{possible groups of five}\end{align*}

**Real-World Application: Student Council**

If your student council has seven members, how many ways are there to reach a majority vote?

There are seven members to choose from, so \begin{align*}n = 7\end{align*}

a. Four votes:

\begin{align*}& \qquad \frac{7!}{(7-4)!\times4!}\\
& \frac{7 \times 6 \times 5 \times 4 {\color{red}\bcancel{ \times 3 \times 2 \times 1}}}{{\color{red}\bcancel{(3 \times 2 \times 1)}}(4 \times 3 \times 2 \times 1)}\\
& \qquad \quad \frac{840}{24}=35\end{align*}

b. Five votes:

\begin{align*}& \quad \frac{7!}{(7-5)! \times (5!)}\\
& \frac{7 \times 6 \times 5 \times 4 \times 3}{5 \times 4 \times 3 \times 2 \times 1}\\
& \quad \ \frac{2520}{120}=21\end{align*}

c. Six votes:

\begin{align*}& \quad \ \frac{7!}{(7-6)! \times 6!}\\
& \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2}{6 \times 5 \times 4 \times 3 \times 2 \times 1}\\
& \qquad \frac{5040}{720}=7\end{align*}

d. Seven votes:

\begin{align*}& \frac{7!}{(7-7)! \times 7!}\\
& \quad \ \ \frac{7!}{7!}=1\end{align*}

**\begin{align*}\text{Total} = 35+21+7+1 = 64\end{align*} Total=35+21+7+1=64 ways to reach a majority vote**

#### Earlier Problem Revisited

Suppose there are fifteen students in a classroom, and you are looking to build a team of five students for a tug-of-war. How many different groups of five students could you create?

This is a combinations problem requiring us to choose items five at a time from a pool of 15, which can be notated as 15C5 (read as "fifteen, choose five"). Use the combinations formula with n=15 and r=5:

\begin{align*}\text{Combinations} & = \frac{n!}{(n-r)! \times r!} \\
& = \frac{ 15! }{ \big( (15-5)! \times 5! \big) } \\
& = \frac{ 15! }{ 10! \times 5! } \\
& = \frac{ 15 \times 14 \times 13 \times 12 \times 11 }{ 5! } \\
& = \frac{ 360,360 }{ 120 }\end{align*}

3,003 different teams

### Examples

#### Example 1

How many ways can six cars be chosen from a lot containing twenty-one cars?

There are twenty-one cars to choose from, \begin{align*}n = 21\end{align*}

There are six cars in each group, \begin{align*}r = 6\end{align*}

\begin{align*}& \qquad \frac{21!}{((21-6)! \times 6!)}\\
& \frac{21 \times 20 \times 19 \times 18 \times 17 \times 16}{6!}\\
& \quad 54264 \ combinations\end{align*}

#### Example 2

How many unique groups of four can be made from nine pieces of candy?

Nine pieces to choose from, \begin{align*}n = 9\end{align*}

\begin{align*}& \frac{9!}{(9-4)! \times 4!}\\
& \frac{9 \times 8 \times 7 \times 6}{4!}\\
& 756 \ groups\end{align*}

#### Example 3

How many unique five-card hands are possible using a standard deck of cards?

There are fifty-two card in a standard deck, \begin{align*}n = 52\end{align*}

\begin{align*}& \quad \quad \ \frac{52!}{(52-5)! \times 5!}\\
& \quad \frac{52 \times 51 \times 50 \times 49 \times 48}{5!}\\
& 62,375,040 \ possible \ hands\end{align*}

### Review

- In the ball closet, there are 6 basketballs, footballs, and baseballs. You go in and pick 2 items at random. Assuming all the balls are marked differently, how many different groups of balls can you end up with?
- You go to a store and buy 4 various items from 6 distinct choices. How many different possibilities were there?
- Find the number of combinations of choosing 5 items from 9 distinct items.
- Find the number of combinations of choosing 6 items from 9 distinct items.
- You have a pile of 7 distinctly different coins. How many ways can you take 2 coins from the pile?
- You are dealt a hand of 4 cards from a deck of 6 cards. Find the number of possible hands you can get
- You hire 2 out of 9 people to build a garden. Find the number of possible ways of choosing your workers.
- Find the number of combinations of choosing 4 items from 7 distinct items.
- Find the number of combinations of choosing 5 items from 5 distinct items.
- How many five-card hands are possible from a deck of twenty-six cards?
- How many unique 7-person combinations are possible using 32 people?
- If Pandi has nineteen chickens in her coop, how many unique groups of six chickens are possible from them?
- If one of her chickens lays colored eggs, every egg unique, how many four-color groups could she make from a dozen eggs?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 8.5.