# 2.6: Combination Problems

Difficulty Level: Basic Created by: CK-12
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Practice Combination Problems

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There are 100 senators in the United States Senate. If they must each enter the entrance to the Senate one-by-one, how many possible ways can this happen? Can you see why a calculator or some other process might be useful to solve this combination problem? Check out an interesting solution at http://puzzles.nigelcoldwell.co.uk/nineteen.htm.

### Watch This

First watch this video to learn about calculating combinations with calculators.

Then watch this video to see some examples.

### Guidance

To calculate combinations (\begin{align*}nCr\end{align*}) on the TI calculator, first enter the \begin{align*}n\end{align*} value, and then press \begin{align*}\boxed{\text{MATH}}\end{align*}. You should see menus across the top of the screen. You want the fourth menu: PRB (arrow right 3 times). The PRB menu should appear as follows:

You will see several options, with \begin{align*}nCr\end{align*} being the third. Press \begin{align*}\boxed{3}\end{align*}, and then enter the \begin{align*}r\end{align*} value. Finally, press \begin{align*}\boxed{\text{ENTER}}\end{align*} to calculate the answer.

#### Example A

Compute \begin{align*}{_{10}}C_6\end{align*} using your TI calculator.

To find the answer, enter the following into your TI calculator:

\begin{align*}\boxed{1} \ \boxed{0} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{6} \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

Therefore, \begin{align*}{_{10}}C_6 &= 210\end{align*}.

#### Example B

Jim has 4 paid vacation days that he has to use before the end of the year. If there are 26 business days left in the year, in how many ways can Jim use his paid vacation days? Compute the answer using your TI calculator.

To find the answer, enter the following into your TI calculator:

\begin{align*}\boxed{2} \ \boxed{6} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{4} \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

Therefore, \begin{align*}{_{26}}C_4 &= 14,950\end{align*}, which means that there are 14,950 ways that Jim can use his paid vacation days.

#### Example C

Kathy has 35 radishes growing in her garden, and she needs to pick 8 to put in the vegetable stew that she's cooking. In how many ways can Kathy pick the 8 radishes? Compute the answer using your TI calculator.

To find the answer, enter the following into your TI calculator:

\begin{align*}\boxed{3} \ \boxed{5} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{8} \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

Therefore, \begin{align*}{_{35}}C_8 &= 23,535,820\end{align*}, which means that there are 23,535,820 ways that Kathy can pick the 8 radishes.

### Guided Practice

If there are 20 rock songs and 20 rap songs to choose from, in how many different ways can you select 12 rock songs and 7 rap songs for a mix CD?

We have multiple groups from which we are required to select, so we have to calculate the possible combinations for each group (rock songs and rap songs in this example) separately and then multiply together.

Using TI technology: for \begin{align*}{_n}C_r\end{align*}, type the \begin{align*}n\end{align*} value (the total number of items), and then press \begin{align*}\boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright}\end{align*} \begin{align*}\text{(PRB)} \ \boxed{\blacktriangledown} \ \boxed{\blacktriangledown}\end{align*} (to number 3) \begin{align*}\boxed{\text{ENTER}}\end{align*}. Then type the \begin{align*}r\end{align*} value (the number of items your want to choose), and finally, press \begin{align*}\fbox{ENTER}\end{align*}.

\begin{align*}\text{Rock:}\end{align*}

For the rock songs, you would enter the following:

\begin{align*}\boxed{2} \ \boxed{0} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{1} \ \boxed{2} \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

Therefore, \begin{align*}{_{20}}C_{12} &= 125,970\end{align*}.

\begin{align*}\text{Rap:}\end{align*}

For the rap songs, you would enter the following:

\begin{align*}\boxed{2} \ \boxed{0} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{7} \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

Therefore, \begin{align*}{_{20}}C_7 &= 77,520\end{align*}.

Thus, the total number of possible combinations is:

\begin{align*}&\text{Rock} \quad \ \text{Rap}\\ &{_{20}}C_{12} \times _{20}C_7 = 125,970 \times 77,520 = 9.765 \times 10^9 \ \text{possible combinations}\end{align*}

This means there are \begin{align*}9.765 \times 10^9\end{align*} different ways can you select 12 rock songs and 7 rap songs for the mix CD.

### Practice

1. Enter each of the following sets of keystrokes into your TI calculator to compute the corresponding combinations.
2. \begin{align*}\boxed{2} \ \boxed{1} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{3} \ \boxed{\text{ENTER}}\end{align*}
3. \begin{align*}\boxed{1} \ \boxed{7} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{7} \ \boxed{\text{ENTER}}\end{align*}
4. \begin{align*}\boxed{3} \ \boxed{0} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{5} \ \boxed{\text{ENTER}}\end{align*}
5. \begin{align*}\boxed{1} \ \boxed{4} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{9} \ \boxed{\text{ENTER}}\end{align*}
6. \begin{align*}\boxed{2} \ \boxed{8} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{4} \ \boxed{\text{ENTER}}\end{align*}
7. \begin{align*}\boxed{1} \ \boxed{9} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{6} \ \boxed{\text{ENTER}}\end{align*}
8. \begin{align*}\boxed{3} \ \boxed{3} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{2} \ \boxed{\text{ENTER}}\end{align*}
9. \begin{align*}\boxed{2} \ \boxed{7} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{8} \ \boxed{\text{ENTER}}\end{align*}
10. \begin{align*}\boxed{2} \ \boxed{0} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{4} \ \boxed{\text{ENTER}}\end{align*}
11. \begin{align*}\boxed{3} \ \boxed{8} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{3} \ (\text{nCr}) \ \boxed{9} \ \boxed{\text{ENTER}}\end{align*}

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### Vocabulary Language: English

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

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