Factorials that involve variables are built the same way as basic factorials. However, you sometimes don't know what the final value is. So there are ways to write these factorials that build on the basic factorials concepts.
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Variable Lengths
The trick when using factorials in calculus is that you may have factorials that look like this.
\(n! = n(n1)(n2). . . 3 \cdot 2 \cdot 1\)
and you don't know what \(n\) is. In this case, the terms on the right are all you can write. However, in most cases, you will have fractions with various factorials in both the numerator and denominator and you will need to simplify. Here is an example.
Example
Simplify \( (n+2)! / n! \)
\(\displaystyle{\frac{(n+2)!}{n!} = }\) \(\displaystyle{ \frac{(n+2)(n+1)(n)(n1). . . 3 \cdot 2 \cdot 1}{(n)(n1). . . 3 \cdot 2 \cdot 1} = }\) \(\displaystyle{ (n+2)(n+1) }\)
There is another way to write this using the fact that factorials build on one another.
\( \begin{array}{rcl} \displaystyle{\frac{(n+2)!}{n!}} & = & \displaystyle{\frac{(n+2)(n+1)(n!)}{n!}} \\ & = & (n+2)(n+1) \end{array} \)
This last way of writing the factorials may be easier to see what cancels.
Things To Watch For
1. Parentheses are very important in factorials. For example, \((2n)! \neq 2n!\) since \(2n! = 2(n!)\).
2. When in doubt, write out the first few terms to make it obvious what is canceling. If it helps, use the idea that factorials build to rewrite factorials.
Time for some practice problems.
Practice
Unless otherwise instructed, simplify these factorials.
\(\displaystyle{\frac{(n3)!}{n!}}\)
Problem Statement 

Simplify \(\displaystyle{\frac{(n3)!}{n!}}\)
Final Answer 

\(\displaystyle{\frac{(n3)!}{n!}}\) \(\displaystyle{ = \frac{1}{(n2)(n1)n}}\)
Problem Statement
Simplify \(\displaystyle{\frac{(n3)!}{n!}}\)
Solution
\(\displaystyle{\frac{(n3)!}{n!}}\) \(\displaystyle{ = \frac{(n3)!}{(n3)! \cdot (n2) \cdot (n1) \cdot n} = \frac{1}{(n2) \cdot (n1) \cdot n} }\)
Final Answer
\(\displaystyle{\frac{(n3)!}{n!}}\) \(\displaystyle{ = \frac{1}{(n2)(n1)n}}\)
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\(\displaystyle{\frac{(2n+1)!}{(2n)!}}\)
Problem Statement 

\(\displaystyle{\frac{(2n+1)!}{(2n)!}}\)
Final Answer 

\(\displaystyle{\frac{(2n+1)!}{(2n)!} = (2n+1)}\)
Problem Statement
\(\displaystyle{\frac{(2n+1)!}{(2n)!}}\)
Solution
\(\begin{array}{rcl} \displaystyle{\frac{(2n+1)!}{(2n)!}} & = & \displaystyle{\frac{1 \cdot 2 \cdot . . . \cdot (2n1) \cdot (2n) \cdot (2n+1)}{1 \cdot 2 \cdot . . . \cdot (2n1) \cdot (2n)}} \\ & = & \displaystyle{\frac{(2n)! \cdot (2n+1)}{(2n)!} = (2n+1)} \end{array} \)
Final Answer
\(\displaystyle{\frac{(2n+1)!}{(2n)!} = (2n+1)}\)
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\(\displaystyle{\frac{(3n1)!}{(3n+1)!}}\)
Problem Statement 

\(\displaystyle{\frac{(3n1)!}{(3n+1)!}}\)
Final Answer 

\(\displaystyle{\frac{(3n1)!}{(3n+1)!} = \frac{1}{(3n)(3n+1)}}\)
Problem Statement
\(\displaystyle{\frac{(3n1)!}{(3n+1)!}}\)
Solution
\(\begin{array}{rcl} \displaystyle{\frac{(3n1)!}{(3n+1)!}} & = & \displaystyle{\frac{(3n1)!}{(3n1)! \cdot (3n) \cdot (3n+1)}} \\ & = & \displaystyle{\frac{1}{(3n)(3n+1)}} \end{array} \)
Final Answer
\(\displaystyle{\frac{(3n1)!}{(3n+1)!} = \frac{1}{(3n)(3n+1)}}\)
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\(\displaystyle{\frac{k!}{(k2)!}}\)
Problem Statement
\(\displaystyle{\frac{k!}{(k2)!}}\)
Solution
video by MIP4U 

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\(\displaystyle{ \frac{(n+2)!}{(n1)!} }\)
Problem Statement 

\(\displaystyle{ \frac{(n+2)!}{(n1)!} }\)
Final Answer 

\( (n+2)(n+1)n = n^3 + 3n^2 + 2 \)
Problem Statement
\(\displaystyle{ \frac{(n+2)!}{(n1)!} }\)
Solution
Check with your instructor to see if they want you to multiply out your answer.
Final Answer
\( (n+2)(n+1)n = n^3 + 3n^2 + 2 \)
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\(\displaystyle{ \frac{(n+1)!}{(n+3)!} }\)
Problem Statement 

\(\displaystyle{ \frac{(n+1)!}{(n+3)!} }\)
Final Answer 

\(\displaystyle{ \frac{1}{(n+3)(n+2)} }\)
Problem Statement
\(\displaystyle{ \frac{(n+1)!}{(n+3)!} }\)
Solution
Check with your instructor to see if they want you to multiply out the denominator.
Final Answer
\(\displaystyle{ \frac{1}{(n+3)(n+2)} }\)
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\(\displaystyle{ \frac{n!}{(n2)!} }\)
Problem Statement 

\(\displaystyle{ \frac{n!}{(n2)!} }\)
Final Answer 

\( n(n1) \)
Problem Statement
\(\displaystyle{ \frac{n!}{(n2)!} }\)
Solution
Check with your instructor to see if they want you to multiply out your answer.
Final Answer
\( n(n1) \)
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\(\displaystyle{ \frac{(2m)!}{(2m+2)!} }\)
Problem Statement 

\(\displaystyle{ \frac{(2m)!}{(2m+2)!} }\)
Final Answer 

\(\displaystyle{ \frac{1}{(2m+2)(2m+1)} }\)
Problem Statement
\(\displaystyle{ \frac{(2m)!}{(2m+2)!} }\)
Solution
Check with your instructor to see if they want you to multiply out the denominator.
Final Answer
\(\displaystyle{ \frac{1}{(2m+2)(2m+1)} }\)
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\(\displaystyle{ \frac{(n+3)!}{n!} }\)
Problem Statement 

\(\displaystyle{ \frac{(n+3)!}{n!} }\)
Final Answer 

\( (n+3)(n+2)(n+1) \)
Problem Statement
\(\displaystyle{ \frac{(n+3)!}{n!} }\)
Solution
Although he says at the end of the problem that you need to foil it out, most instructors will not require you to do that for these types of problems.
Final Answer
\( (n+3)(n+2)(n+1) \)
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\(\displaystyle{ \frac{(3n+2)!}{(3n1)!} }\)
Problem Statement 

\(\displaystyle{ \frac{(3n+2)!}{(3n1)!} }\)
Final Answer 

\( (3n+2)(3n+1)(3n) \)
Problem Statement
\(\displaystyle{ \frac{(3n+2)!}{(3n1)!} }\)
Solution
Again, do not multiply out the answer unless specifically told to do so by your instructor.
Final Answer
\( (3n+2)(3n+1)(3n) \)
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\(\displaystyle{ \frac{(n^24)!}{(n2)(n^25)!} }\)
Problem Statement 

\(\displaystyle{ \frac{(n^24)!}{(n2)(n^25)!} }\)
Hint 

The last factorial in the denominator applies only to the \((n^25)\) term. So the denominator can be written more clearly as \((n2)[(n^25)!]\).
Problem Statement 

\(\displaystyle{ \frac{(n^24)!}{(n2)(n^25)!} }\)
Final Answer 

\(\displaystyle{ \frac{n^24}{n2} }\)
Problem Statement
\(\displaystyle{ \frac{(n^24)!}{(n2)(n^25)!} }\)
Hint
The last factorial in the denominator applies only to the \((n^25)\) term. So the denominator can be written more clearly as \((n2)[(n^25)!]\).
Solution
Although they give the final answer as \(n+2\), you know from the domain and range page that they should have also said that \( n \neq 2 \) as part of the final answer.
Final Answer
\(\displaystyle{ \frac{n^24}{n2} }\)
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\(\displaystyle{ \frac{(n!)^2}{(n1)!(n+1)!} }\)
Problem Statement 

\(\displaystyle{ \frac{(n!)^2}{(n1)!(n+1)!} }\)
Final Answer 

\(\displaystyle{ \frac{n}{n+1} }\)
Problem Statement
\(\displaystyle{ \frac{(n!)^2}{(n1)!(n+1)!} }\)
Solution
Final Answer
\(\displaystyle{ \frac{n}{n+1} }\)
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\(\displaystyle{ \frac{(n+3)!n!}{(n+2)!(n1)!} }\)
Problem Statement 

\(\displaystyle{ \frac{(n+3)!n!}{(n+2)!(n1)!} }\)
Final Answer 

\(n(n+3)\)
Problem Statement
\(\displaystyle{ \frac{(n+3)!n!}{(n+2)!(n1)!} }\)
Solution
Check with your instructor to see if they want you to multiply out your answer.
Final Answer
\(n(n+3)\)
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\(\displaystyle{ \frac{(n+2)!n!}{(n+1)!} }\)
Problem Statement 

\(\displaystyle{ \frac{(n+2)!n!}{(n+1)!} }\)
Final Answer 

\( n+2  1/(n+1) \)
Problem Statement
\(\displaystyle{ \frac{(n+2)!n!}{(n+1)!} }\)
Solution
Final Answer
\( n+2  1/(n+1) \)
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\(\displaystyle{\frac{(k+2)!}{(k1)!}}\)
Problem Statement
\(\displaystyle{\frac{(k+2)!}{(k1)!}}\)
Solution
This problem is solved in two separate videos by two different instructors. Watching both of them will help you better understand these concepts.
video by MIP4U 

video by kbrescher 

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