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# Fundamental Counting Principle with and without Repetition

## Determine sample spaces using the principle of counting

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Using the Fundamental Counting Principle with and without Repetition

A lock has the digits 0-39. A series of three numbers unlocks the lock. How many possible unlocking combinations are there if the numbers cannot be repeated?

### Fundamental Counting Principle

Consider a phone number. A phone number consists entirely of numbers that might be repeated. In this concept we will look at how to determine the total number of possible combinations of items which may be repeated.

Let's solve the following problems.

1. A license plate consists of three letters and four numbers in the state of Virginia. If letters and numbers can be repeated, how many possible license plates can be made?

If we consider the three slots for the letters, how many letters can be chosen to place in each slot? How about the four slots for the numbers? If there are no restrictions, i.e. letter and numbers can be repeated, the total number of license plates is:

26×26×26×10×10×10×10=175,760,000\begin{align*}\underline{{\color{red}26}} \times \underline{{\color{red}26}} \times \underline{{\color{red}26}} \times \underline{{\color{blue}10}} \times \underline{{\color{blue}10}} \times \underline{{\color{blue}10}} \times \underline{{\color{blue}10}}=175,760,000\end{align*}

Now, what if letters or numbers could not be repeated? Well, after the first letter is chosen, how many letters could fill the next spot? Since we started with 26, there would be 25 unused letters for the second slot and 24 for the third slot. Similarly with the numbers, there would be one less each time:

26×25×24×10×9×8×7=78,624,000\begin{align*}\underline{{\color{red}26}} \times \underline{{\color{red}25}} \times \underline{{\color{red}24}} \times \underline{{\color{blue}10}} \times \underline{{\color{blue}9}} \times \underline{{\color{blue}8}} \times \underline{{\color{blue}7}}=78,624,000\end{align*}

1. How many unique five letter passwords can be made? How many can be made if no letter is to be repeated?

Since there are 26 letters from which to choose for each of 5 slots, the number of unique passwords can be found by multiplying 26 by itself 5 times or (26)5=11,881,376\begin{align*}(26)^5=11,881,376\end{align*}. If we do not repeat letters, then we need to subtract one each time we multiply: 26×25×24×23×22=7,893,600\begin{align*}26 \times 25 \times 24 \times 23 \times 22=7,893,600\end{align*}.

1. How many unique 4 digit numbers can be made? What if no digits can be repeated?

For the first part, consider that in order for the number to be a four digit number, the first digit cannot be zero. So, we start with only 9 digits for the first slot. The second slot could be filled with any of the ten digits and so on:

9×10×10×10=9000.\begin{align*}\underline{9} \times \underline{10} \times \underline{10} \times \underline{10}=9000.\end{align*}

For the second part, in which digits cannot be repeated, we would still have 9 possible digits for the first slot, then we’d have 9 again for the second slot (we cannot repeat the first digit, but we can add 0 back into the mix), then 8 for the third slot and 7 for the final slot:

9×9×8×7=4536.\begin{align*}\underline{9} \times \underline{9} \times \underline{8} \times \underline{7}=4536.\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find the number of possible unlocking combinations if the numbers cannot be repeated.

Since there are 40 numbers from which to choose for each of 3 slots, the number of unique passwords can be found by multiplying 40 by itself 3 times or (40)3=64,000\begin{align*}(40)^3=64,000\end{align*}. However, we cannot repeat numbers so we need to subtract one each time we multiply: 40×39×38=59,280\begin{align*}40 \times 39 \times 38 =59,280\end{align*}.

Therefore, there are 59,280 possible unlocking combinations.

#### Example 2

How many unique passwords can be made from 6 letters followed by 1 number or symbol if there are ten possible symbols? No letters or numbers can be repeated.

26×25×24×23×22×21×20=3,315,312,000\begin{align*}\underline{{\color{red}26}} \times \underline{{\color{red}25}} \times \underline{{\color{red}24}} \times \underline{{\color{red}23}} \times \underline{{\color{red}22}} \times \underline{{\color{red}21}} \times \underline{{\color{blue}20}}=3,315,312,000\end{align*}

#### Example 3

If a license plate has three letters and three numbers, how many possible combinations can be made?

26×26×26×10×10×10=17,576,000\begin{align*}\underline{{\color{red}26}} \times \underline{{\color{red}26}} \times \underline{{\color{red}26}} \times \underline{{\color{blue}10}} \times \underline{{\color{blue}10}} \times \underline{{\color{blue}10}} =17,576,000\end{align*}

#### Example 4

In a seven digit phone number, the first three digits represent the exchange. If, within a particular area code, there are 53 exchanges, how many phone numbers can be made

53×10×10×10×10=530,000\begin{align*}\underline{{\color{red}53}} \times \underline{{\color{blue}10}} \times \underline{{\color{blue}10}} \times \underline{{\color{blue}10}} \times \underline{{\color{blue}10}}=530,000\end{align*}

### Review

Use the Fundamental Counting Principle to answer the following questions. Refer back to the examples and guided practice for help.

1. How many six digit numbers can be formed if no digits can be repeated?
2. How many five digit numbers can be formed that end in 5?
3. How many license plates can be formed of 4 letters followed by 2 numbers?
4. How many seven digit phone numbers can be made if there are 75 exchanges in the area?
5. How many four letter pins (codes) can be made?
6. How many four number/letter pins can be made if no number or letter can be repeated?
7. How many different ways can nine unique novels be arranged on a shelf?
8. How many different three scoop cones can be made from 12 flavors of ice cream allowing for repetition? What if no flavors can be repeated?
9. How many different driver’s license numbers can be formed by 2 letters followed by 6 numbers?
10. How many student ID numbers can be made by 4 random digits (zero cannot come first) followed by the student’s grade (9, 10, 11 or 12). Example: 5422-12 for a 12th\begin{align*}12^{th}\end{align*} grader.

To see the Review answers, open this PDF file and look for section 12.2.

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