Question

How do you simplify the factorial expression $ \dfrac{{(2n - 1)!}}{{(2n + 1)!}}? $

Answer 1

Hint: Separately write the factorial values of numerator and denominator, then take the common factor between them out or cancelled. This is a much similar process to simplifying fraction numbers.
For your knowledge, factorial of a number represents the product of the numbers up to the given number, it is written mathematically as follows
 $ n! = 1 \times 2 \times 3 \times 4 \times ...... \times (n - 1) \times n $

Complete step-by-step answer:
In order to simplify a factorial expression, we first have to expand factorials given in the expression and then take out the common factors between them and perform algebraic operations according to the problem.
Coming to the problem, we have to simplify the factorial expression $ \dfrac{{(2n - 1)!}}{{(2n + 1)!}} $ , so first writing the factorial values of the numerator and the denominator,
 $ \Rightarrow \dfrac{{(2n - 1)!}}{{(2n + 1)!}} = \dfrac{{1 \times 2 \times 3 \times .... \times n \times (n + 1) \times ....(2n - 2) \times (2n - 1)}}{{1 \times 2 \times 3 \times .... \times n \times (n + 1) \times ....(2n - 2) \times (2n - 1) \times 2n \times (2n + 1)}} $
Now cancelling out the common factors or terms between the numerator and the denominator, we will get
 $
   \Rightarrow \dfrac{{(2n - 1)!}}{{(2n + 1)!}} = \dfrac{{1 \times 2 \times 3 \times .... \times n \times (n + 1) \times ....(2n - 2) \times (2n - 1)}}{{1 \times 2 \times 3 \times .... \times n \times (n + 1) \times ....(2n - 2) \times (2n - 1) \times 2n \times (2n + 1)}} \\
   \Rightarrow \dfrac{{(2n - 1)!}}{{(2n + 1)!}} = \dfrac{1}{{2n \times (2n + 1)}} \;
  $
Therefore $ \dfrac{1}{{2n \times (2n + 1)}} $ is the simplified form of the given factorial expression $ \dfrac{{(2n - 1)!}}{{(2n + 1)!}} $
So, the correct answer is “$ \dfrac{{(2n - 1)!}}{{(2n + 1)!}} $”.

Note: Factorial is a function that multiplies a number with each and every number below it up to one. Mathematically it can be understood as follows
 $ n! = = 1 \times 2 \times 3 \times 4 \times ...... \times (n - 1) \times n $
 $ n! $ is the new notation to represent “n” factorial, there was an old one too and some people do use that old notation in which “n” factorial is represented as $ \left| \!{\underline {\,
  n \,}} \right. $
Factorial functions are normally used in the evaluation of Permutation and Combination (P & C) and to determine the value of coefficients of the binomial expansions.
Factorial function has a domain of all positive integers, also the value of $ 0! = 1 $ and also it has a range of positive integers.