Division of Factorials: Definition & Concept

Division of Factorials: Definition & Concept
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  • 0:00 Notation And Function…
  • 1:02 Division Of Factorials
  • 1:23 Using The Factorial Function
  • 2:25 Factorial Divisions In…
  • 4:00 Factorial Combination…
  • 6:39 Lesson Summary
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Lesson Transcript
Instructor: David Liano
After completing this lesson, you will know how to use the factorial function and factorial notation. You will also be able to use factorials in problem solving, which often requires the division of factorials.

Notation and Function for Factorials

The factorial function uses a special symbol, !, and is usually shown like this, n!. The domain of n is the set of natural numbers. In other words, n can be any natural number.

The factorial function n! is the product of all the natural numbers from 1 to n. In symbols, we can show the function as n! = n * (n - 1) * (n - 2) *, . . ., 2 * 1. It is usually written in ascending or descending order, but this lesson will usually write the factors of a factorial in descending order.

Let's look at an example.

4! = 4 * 3 * 2 * 1

Zero is usually not included in the set of natural numbers, but 0! might appear in some problems. 0! is simply defined as follows:

0! = 1

Division of Factorials

The division of factorials is exactly what it states. It is a division problem with factorials in the numerator and/or denominator. For example, the following expression is a division of factorials:

6! / 4!

We will solve this problem in an example that comes later in this lesson. Let's first look at a common way for using factorials.

Using the Factorial Function

Factorial functions are useful for determining how many ways a set of objects can be arranged. The ordering of a set number of objects is called a permutation. Let's say that we have 6 different books and want to determine how many ways we can arrange these books on a single shelf.

There are 6 choices for the first spot. For the second spot, 5 choices remain. Therefore, each book that could be in the first spot can be followed by any of the 5 books that remain or 6 * 5. Then there are 4 choices for the third spot, so 6 * 5 * 4. This pattern continues until all the books are arranged. There are 6 * 5 * 4 * 3 * 2 * 1 = 720 ways to arrange the 6 different books.

Arrangement of Six Books
six books

This example gives us one explanation for why 0! = 1. How many ways are there to arrange zero objects? There is one way, which is the empty set. Think of an empty bookshelf. That would be how many ways we can arrange zero books.

Factorial Division in Permutations

Let's look at the book example again. What if we wanted to arrange only 2 of the 6 books on the shelf? The typical formula for arranging k objects from a group of n distinct objects is shown in the figure on screen now:

Figure 1
permutation

Let's use this formula for our example. We have 6 different books, so n = 6. We arrange only 2 of the books, so k = 2. Let's plug these values into our formula:

6! / (6 - 2)! = 6! / 4!

If we write out all the factors of each factorial, we get the following:

(6 * 5 * 4 * 3 * 2 * 1) / (4 * 3 * 2 * 1)

We can cancel out 4 * 3 * 2 * 1 from the numerator and denominator and are left with 6 * 5, so our final answer is 30. In other words, we cancel out 4!. Let's look at it another way. We only want to arrange two books, so we need to eliminate 4!, which represents the placement of the four remaining books. It should now be apparent that the factorial of a natural number is a subset of a factorial of any greater natural number.

Let's go back to our first example, in which we arranged all 6 books. If we use our formula in Figure 1, we obtain the following:

6! / (6 - 6)! = 6! / 0! = 720 / 1 = 720

This shows why it is handy for 0! to equal 1.

Factorial Combination in Combinations

Factorial functions are also useful in grouping a certain number of objects when the arrangement or order is not important. These unordered groupings are called combinations.

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