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Question

Question: How do you simplify \(\dfrac{(2n)!}{n!}\)?...

How do you simplify (2n)!n!\dfrac{(2n)!}{n!}?

Explanation

Solution

A factorial of a number multiplied by all of the integers preceding it. Example: “Five Factorial” = 5!=5×4×3×2×1=1205!=5\times 4\times 3\times 2\times 1=120. Hence factorial of a number nn is the product of all the numbers from 11 to nn. So we have
n!=n⋅(n−1)⋅(n−2)⋅...⋅2⋅1n!=n\cdot (n-1)\cdot (n-2)\cdot ...\cdot 2\cdot 1
Although there is no simplification of (2n)!n!\dfrac{(2n)!}{n!}, but there are other ways of expressing it, such as
(2n)!n!=∏k=0n−1(2n−k)=(2n)(2n−1)...(n+1)\dfrac{(2n)!}{n!}=\prod\limits_{k=0}^{n-1}{(2n-k)=(2n)(2n-1)...(n+1)} .

Complete step by step solution:
To simplify: (2n)!n!\dfrac{(2n)!}{n!}
By the definition of factorial function,
(2n)!=(2n)(2n−1)...2⋅1(2n)!=(2n)(2n-1)...2\cdot 1
And n!=n(n−1)(n−2)...⋅2⋅1n!=n(n-1)(n-2)...\cdot 2\cdot 1
Now putting the value of (2n)!(2n)! in (2n)!n!\dfrac{(2n)!}{n!} , we get
(2n)!n!=1n!(1⋅2⋅3⋅...⋅2n)\dfrac{(2n)!}{n!}=\dfrac{1}{n!}(1\cdot 2\cdot 3\cdot ...\cdot 2n)
This can also be written as:
(2n)!n!=1n!(2⋅4⋅6⋅...⋅2n)(1⋅3⋅5⋅...⋅(2n−1)) =1n!(1⋅2)(2⋅2)(3⋅2)...(n⋅2)(1⋅3⋅5⋅...(2n−1)) =1n!(1⋅2⋅3⋅...n)2n(1⋅3⋅5⋅...⋅(2n−1)) =1n!n!⋅2n(1⋅3⋅5⋅...⋅(2n−1)) =2n(1⋅3⋅5⋅...⋅(2n−1)) \begin{aligned} & \dfrac{(2n)!}{n!}=\dfrac{1}{n!}(2\cdot 4\cdot 6\cdot ...\cdot 2n)(1\cdot 3\cdot 5\cdot ...\cdot (2n-1)) \\\ & =\dfrac{1}{n!}(1\cdot 2)(2\cdot 2)(3\cdot 2)...(n\cdot 2)(1\cdot 3\cdot 5\cdot ...(2n-1)) \\\ & =\dfrac{1}{n!}(1\cdot 2\cdot 3\cdot ...n){{2}^{n}}(1\cdot 3\cdot 5\cdot ...\cdot (2n-1)) \\\ & =\dfrac{1}{n!}n!\cdot {{2}^{n}}(1\cdot 3\cdot 5\cdot ...\cdot (2n-1)) \\\ & ={{2}^{n}}(1\cdot 3\cdot 5\cdot ...\cdot (2n-1)) \\\ \end{aligned}

Therefore, (2n)!n!=2n(1⋅3⋅5⋅...⋅(2n−1))\dfrac{(2n)!}{n!}={{2}^{n}}(1\cdot 3\cdot 5\cdot ...\cdot (2n-1)).

Note: Once we know the value of n which can be either positive, negative, or zero, we can easily find the value of (2n)!n!\dfrac{(2n)!}{n!} by putting n in the solution. Factorials have a usage in various areas of mathematics such as probability, combinations and permutations, Taylor’s series, exponential functions, Binomial expansion, and many more.