You're having a party at your house this weekend and you've stocked up on soft drinks. You have five cans of lemon-lime, seven of cola, and eight of root beer. If your friends come to the frig and take them out in succession in how many different orders can that happen?

### Watch This

First watch this video to learn about permutations with repetition.

CK-12 Foundation: Chapter2PermutationswithRepetitionA

Then watch this video to see some examples.

CK-12 Foundation: Chapter2PermutationswithRepetitionB

### Guidance

There is a subset of permutations that takes into account that there are double objects or **repetitions** in a permutation problem. In general, repetitions are taken care of by dividing the permutation by the factorial of the number of objects that are identical.

#### Example A

If you look at the word TOOTH, there are 2 O’s in the word. Both O’s are identical, and it does not matter in which order we write these 2 O’s, since they are the same. In other words, if we exchange 'O' for 'O', we still spell TOOTH. The same is true for the T’s, since there are 2 T’s in the word TOOTH as well. In how many ways can we arrange the letters in the word TOOTH?

We must account for the fact that these 2 O’s are identical and that the 2 T’s are identical. We do this using the formula:

\begin{align*}\frac{_nP_r}{x_1! x_2!}\end{align*}, where \begin{align*}x\end{align*} is the number of times a letter is repeated.

\begin{align*}\frac{_nP_r}{x_1! x_2!} &= \frac{_5P_5}{2!2!}\\ \frac{_5P_5}{2!2!} &= \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1}\\ \frac{_5P_5}{2!2!} &= \frac{120}{4}\\ \frac{_5P_5}{2!2!} &= 30\end{align*}

We can arrange the letters in the word TOOTH in 30 different orders.

#### Example B

How many different 5-letter arrangements can be formed from the word APPLE?

There are 5 letters in the word APPLE, so \begin{align*}n = 5\end{align*}. We want 5-letter arrangements; therefore, we are choosing 5 objects at a time. In this example, \begin{align*}r = 5\end{align*}, and we are using a word with letters that repeat. In the word APPLE, there are 2 P’s, so \begin{align*}x_1 = 2\end{align*}.

There are 60 5-letter arrangements that can be formed from the word APPLE.

#### Example C

How many different 6-digit numerals can be written using the following 7 digits? Assume the repeated digits are all used.

3, 3, 4, 4, 4, 5, 6

There are 7 digits, so \begin{align*}n = 7\end{align*}. We want 6-digit arrangements; therefore, we are choosing 6 objects at a time. In this example, \begin{align*}r = 6\end{align*}, and we are using a group of digits with numbers that repeat. In the group of 7 digits (3, 3, 4, 4, 4, 5, 6), there are two 3’s and three 4’s, so \begin{align*}x_1 = 2\end{align*} and \begin{align*}x_2 = 3\end{align*}.

There are 420 6-digit numerals that can be written using the 7 digits.

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### Guided Practice

If there are 4 chocolate chip, 2 oatmeal, and 2 double chocolate cookies in a box, in how many different orders is it possible to eat all of these cookies?

**Answer:**

There are 8 cookies, so \begin{align*}n = 8\end{align*}. We want to eat all of the cookies; therefore, we are choosing 8 objects at a time. In this example, \begin{align*}r = 8\end{align*}, and we are eating a group of cookies with flavors that repeat. In the group of 8 cookies, there are 4 chocolate chip, 2 oatmeal, and 2 double chocolate cookies, so \begin{align*}x_1 = 4\end{align*}, \begin{align*}x_2 = 2\end{align*}, and \begin{align*}x_3 = 2\end{align*}.

It is possible to eat the cookies in 420 different orders.

### Explore More

- In how many ways can the letters of the word REFERENCE be arranged?
- In how many ways can the letters of the word MISSISSIPPI be arranged?
- In how many ways can the letters of the word MATHEMATICS be arranged?
- A math test is made up of 15 multiple choice questions. 5 questions have the answer A, 4 have the answer B, 3 have the answer C, 2 have the answer D, and 1 has the answer E. How many answer sheets are possible?
- How many different 5-digit numerals can be written using the following 9 digits? 2, 2, 2, 7, 7, 8, 8, 8, 9
- How many different 4-digit numerals can be written using the following 10 digits? 1, 3, 3, 4, 4, 5, 5, 6, 6, 9
- How many different 6-digit numerals can be written using the following 12 digits? 1, 1, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9
- If there are 4 cans of cola, 3 cans of lemonade, and 5 cans of iced tea in a cooler, in how many orders is it possible to consume these drinks?
- A clothing store has a certain shirt in 4 sizes: small, medium, large, and extra large. If it has 2 small, 3 medium, 6 large, and 2 extra large in stock, in how many orders can it sell all the shirts?
- In question 9, suppose the store decides not to sell the extra large shirts. In how many orders can it sell the remaining shirts?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 2.4.