Question

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

Answer 1

We know that $0! = 1$ . The factorial operation, n! is defined as $n! = n(n - 1)(n - 2).......(3)(2)(1)$. In other words, you multiply the number n, being operated upon, by every positive integer smaller than n. Thus, by using this, we will substitute the formula of n! and (n+2)! . After evaluating this, we will get the final output.

Complete step by step answer:
Sets of elements have special operations used to combine them or change them. Another operation that’s used with sets (but that isn’t exclusive to sets) is factorial, denoted by the exclamation point.In short, factorials are products, indicated by an exclamation mark.

Given that, $\dfrac{{\left( {n + 2} \right)!}}{{n!}}$. We have to simplify the factorial expression given in the fractional form. Now, we use the definition of factorial to see how we can expand the factorials of both numerator and denominator. The factorial of n:
$n! = n(n - 1)(n - 2).......(3)(2)(1)$

We can see that factorial of a number is product of all integers starting from 1 to that number so we can expand the given values in the numerator as:
The factorial of (n+2) will be:
$(n + 2)! = (n + 2)(n + 1)n(n - 1)(n - 2).......(3)(2)(1)$
$\Rightarrow (n + 2)!= (n + 2)(n + 1)n!$

Now, we will use this to solve the given expression as below:
$\dfrac{{\left( {n + 2} \right)!}}{{n!}}$
Substitute the given values, we will get,
$\dfrac{{\left( {n + 2} \right)!}}{{n!}} = \dfrac{{(n + 2)(n + 1)n!}}{{n!}}$
We know that the $n! = n(n - 1)(n - 2).......(3)(2)(1)$ and so using this, we will get,
$\dfrac{{\left( {n + 2} \right)!}}{{n!}} = \dfrac{{(n + 2)(n + 1)n(n - 1)(n - 2).......(3)(2)(1)}}{{n(n - 1)(n - 2).......(3)(2)(1)}}$
Now we will cancel the similar terms in both the numerator and denominator, we will get,
$\therefore \dfrac{{\left( {n + 2} \right)!}}{{n!}} = (n + 2)(n + 1)$

Hence, the value of $\dfrac{{\left( {n + 2} \right)!}}{{n!}} = (n + 2)(n + 1)$.

Note: We use the factorial operation in the formulas used to count the number of elements in the union, intersection, or complement of sets. We can check our answer by putting a value of n in the expression and solve it then we check if it is the same when the substitution is made in the result expression. Factorials can be represented in two ways by putting an exclamation mark (!) after an expression or by putting the number in an ‘L’ shaped symbol. Both the notations are correct and acceptable by the math fraternity.