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How to Multiply Factorials

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Reviewed by Joseph Meyer

Last Updated: January 12, 2024 Fact Checked

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Factorials, denoted by a $!$ sign, are products of a whole number and all of the whole numbers below it. It is easy to calculate and multiply two factorials using a scientific calculator’s $x!$ function. You can also multiply factorials by hand. The easiest way to do it is to calculate each factorial individually, and then multiply their products together. You can also use certain rules of factorials to pull out common factors, which can simplify the multiplication process.

Method 1

Method 1 of 3:

Understanding Factorials

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1
Identify a factorial. A factorial, denoted by a whole number with an exclamation point, is the product of a series of sequential whole numbers. ^{[1]
X
Research source}
- For example, $6!$ is a factorial.
- Keep in mind that the factorial of 0 (0!) is 1 by convention. There also isn't any definition for the factorial of negative numbers—it's only possible to calculate the factorial of 0 and other positive numbers.
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2
Evaluate a factorial using a formula. The formula is $n!=n(n-1)(n-2)\cdot \cdot \cdot 3\cdot 2\cdot 1$ . ^{[2]
X
Research source} This means that you extend the sequence of numbers until you get to 1.
- For example, $6!=6(6-1)(6-2)(6-3)(6-4)(6-5)=6(5)(4)(3)(2)(1)$
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3
Calculate a factorial. To calculate a factorial, begin with the denoted number, and multiply it by each sequential whole number, down to 1. ^{[3]
X
Research source} A quick way to calculate a factorial is to use the $x!$ key on a scientific calculator. First hit the number, then hit the $x!$ key to see the product. ^{[4]
X
Research source}
- For example, $6!=6\times 5\times 4\times 3\times 2\times 1=720$ .

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Method 2

Method 2 of 3:

Calculating the Factorials Separately

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1
Calculate the first factorial. Use a calculator for larger numbers. If calculating by hand, make sure you multiply each sequential number, down to 1. Rewrite the equation with this product in parentheses as the first factor. ^{[5]
X
Research source}
- For example, if you are calculating $5!\times 7!$ , first calculate $5!$ :
  $5!\times 7!$
  $=(5\times 4\times 3\times 2\times 1)\times (7!)$
  $=(120)\times (7!)$
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2
Calculate the second factorial. You can do this by calculator or by hand, beginning on the complexity of the factorial. Rewrite the equation with this product as the second factor. ^{[6]
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Research source}
- For example:
  $(120)\times (7!)$
  $=(120)\times (7\times 6\times 5\times 4\times 3\times 2\times 1)$
  $=(120)\times (5040)$
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3
Multiply the two factorials’ products. This will give you the product of the two factorials. Since factorials tend to be large numbers, using a calculator will make this calculation easier. ^{[7]
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Research source}
- For example, $(120)\times (5040)=604,800$ . So, $5!\times 7!=604,800$ .

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Method 3

Method 3 of 3:

Finding Common Factors

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1
Use a formula to factor out the largest common factorial. The formula is $n!=n\times (n-1)!$ . ^{[8]
X
Research source} This means that a smaller factorial is a factor of a larger factorial. ^{[9]
X
Research source} For example, $4!=4\times (4-1)!=4\times 3!$ . When you are multiplying two factorials, the largest common factorial is the smaller of the two factorials.
- For example, if you are calculating $5!\times 7!$ , you can factor out $5!$ from $7!$ :
  $5!\times 5!(7\times 6)$
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2
Rewrite the equation, showing the common factorial as a squared value. Then, calculate the factorial and square its product. ^{[10]
X
Research source}
- For example,
  $5!\times 5!(7\times 6)$
  $=7\times 6\times (5!)^{{2}}$
  $=42\times (120)^{{2}}$
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3
Multiply the remaining factors. The result will be the product of the two original factorials. ^{[11]
X
Research source}
- For example:
  $42\times (120)^{{2}}$
  $=42\times 14,400$
  $=604,800$

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Community Q&A

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Question

How do I multiply two factorials so that the end product is also a factorial? I.e. 3! X 5!

Donagan

Top Answerer

There is no general rule covering this situation. In your example, however, (3!)(5!) = 720 = 6!.

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Question

What is multiplication of any number by 2 factorial?

Donagan

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Multiplying by 2! means multiplying by 2.

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Question

How do I answer this one? (k+1)! + (k+1)!

Donagan

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Multiply (k+1)! by two.

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About This Article

Reviewed by:

Math Teacher

This article was reviewed by Joseph Meyer . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 117,028 times.

31 votes - 58%

Co-authors: 8

Updated: January 12, 2024

Views: 117,028

Categories: Multiplication and Division

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Thanks to all authors for creating a page that has been read 117,028 times.

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