What Is a Factorial?
How a Factorial Works
A factorial is derived by multiplication. Given any whole number (not a fraction), the factorial can be found by multiplying all the whole numbers together from the given down to one. Here are two quick examples:
- If three is the given whole number, then three factorial is 3 x 2 x 1 or 6.
- If six is the given whole number, then six factorial is 6 x 5 x 4 x 3 x 2 x 1 or 720.
A factorial is designated by an exclamation point (!), so a three factorial would be designated 3!. Factorials can be used to find the number of different ways a set of objects can be arranged. Here is an example with explanation:
Given a set of letters (H,U,B,P,A,G,E,S), how many ways can it be rearranged?
- Find the number of individual objects in the set. In this case there are 8 objects.
- Look at the illustration below. Each box is a placeholder representing one position an object could occupy.
- Start at the box furthest to the left. How many different choices are available to occupy that box? The answer is eight. The given set has eight objects. There are eight possibilities so far.
- Moving one box to the right, how many different choices are available to occupy that box? The answer is seven. The given set has eight objects, but 1 has been used in the first box (8-1=7). There are now 8 x 7 or 56 possibilities.
- Moving one more box to the right, how many different choices are available to occupy that box? The answer is six. The given set has eight objects, but two of them have been used to populate the first two boxes (8-2=6). There are now 8 x 7 x 6 or 336 possibilities.
- Following the same sequence of logic, when the furthest box to the right is filled, there will be only one choice left from the original set. The given set has eight objects, but seven of them have been used to populate the first seven boxes (8-7=1). There are now 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 or 40,320 possibilites. That is 8!
Comments or Questions?
I hope this was informative, and you have a better understandig of factorials. If you have a calculator, check for the factorial designation on one of the keys. It is usually shown as X!. Now you know what it is used for.
If you have any comments or questions, please let me know.