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College Algebra: Tutoring Solution11 chapters | 84 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we learn how to find 5 factorial. We talk about different areas of mathematics in which factorials are used, and we also look at some applications that involve factorials and 5 factorial, specifically.

In mathematics, an exclamation mark (!) is used to represent factorials. In general, *n*! represents *n* factorial, and it means that we want to multiply all the integers from *n* down to 1 together. The image shows this formula.

We use the notation 5! to represent 5 factorial. To find 5 factorial, or 5!, simply use the formula; that is, multiply all the integers together from 5 down to 1.

5! = 5 * 4 * 3 * 2 * 1 = 120

When we use the formula to find 5!, we get 120. So, 5! = 120.

Factorials show up in many areas of mathematics, such as statistics, probability, calculus, and trigonometry. They're actually pretty well known for their use in **combinatorics**, which is a fancy name for counting techniques. Combinations and permutations make up a large part of combinatorics, and factorials are an essential part of both of these. Defined simply, combinations and permutations are arrangements of objects. In combinations, order does not matter, and in permutations, order does matter.

For example, if we're talking about 4-digit lock codes, the code 1234 is different than the code 4321, even though it contains the same numbers. Thus, order matters, and the lock codes are permutations of four digits. On the other hand, if we are putting a salad together with the ingredients lettuce, tomato, chicken, and onions, it is the same salad if we list the ingredients as tomato, lettuce, onions, and chicken. In this case, order does not matter, so the salad is a combination of four ingredients.

Factorials come into play when we're talking about formulas that give us the number of permutations or combinations of a number of objects. The images on your screen show some combination and permutation formulas. You don't have to memorize all of these formulas for the purpose of this lesson, but you should pause for a moment and take a closer look at how factorials are intricately used.

Suppose we want to know how many ways there are for five people to finish a race. People can place first, second, third, fourth, or fifth. If the names of the competitors are Alex, Kennedy, Brock, Abrianna, and Peyton, consider them finishing in the following two orders:

Kennedy, Brock, Alex, Abrianna, Peyton

Brock, Alex, Kennedy, Peyton, Abrianna

We see that these represent two different possible ways for them to finish the race. Order matters, so we are working with permutations. Furthermore, a person can't finish in more than one place (for instance, a person can't come in first and second place), so repetition isn't allowed. Therefore, we want to use the formula for permutations with no repetition allowed, or *n*!. In our example, there are five competitors, so we want to find 5!, which we already found to be 120. This tells us there are 120 possible ways for the competitors to finish the race.

Let's consider another scenario. Suppose you want to make a salad with five different ingredients. You have seven ingredients to choose from, and repetition is not allowed. In other words, you can't have the same ingredient count as two different ingredients. As we explained earlier, the order in which you add the ingredients makes no difference, they still make the same five-ingredient salad, so we're working with combinations. We want to use the formula for a combination of *r* objects from *n* objects, and repetition is not allowed. In our scenario, *n* = 7 and *r* = 5. We plug these into the appropriate formula, and simplify as shown here:

We see here that, after dividing 5,040 by 240, that there are 21 possible 5-ingredient salads we can make from 7 ingredients. We also see that 5! appeared in this application as well as the last.

Let's now take a moment or two to review the important information that we've learned while discovering how to solve 5!. The first important thing we learned is that an exclamation mark (!) is the symbol used to denote a factorial, which is when you multiply all of the integers together from the origin, as in the number next to the exclamation point all the way down to one, to get your answer. In the case of 5!, it's:

5 * 4 * 3 * 2 * 1 = 120

Factorials show up often in many areas of mathematics, such as their use in **cominatorics**, which is a fancy name for counting techniques. This ubiquity is why it's extremely useful to know how to calculate and work with them.

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College Algebra: Tutoring Solution11 chapters | 84 lessons

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