Question

Find the value of

1. $8!$
2. $4! - 3!$

Answer 1

Here we will use the concept of factorial to solve the question. Factorial of a number is termed as the multiplication of all natural numbers equal to or less than that number. It is denoted by an exclamation mark ($!$). We will use the formula of factorial to evaluate the given factorials.
Formula Used: Here, we have used the formula of factorial of a number ${\rm{n}}$, ${\rm{n}}! = {\rm{n}} \times \left( {{\rm{n}} - 1} \right) \times \left( {{\rm{n}} - 2} \right) \times .... \times 2 \times 1$.

Complete step by step solution:

1. We know that the formula of factorial of a number is ${\rm{n}}! = {\rm{n}} \times \left( {{\rm{n}} - 1} \right) \times \left( {{\rm{n}} - 2} \right) \times .... \times 2 \times 1$.
   So, to find the value of $8!$ we will substitute ${\rm{n}} = 8$ in the formula.
   $8! = 8 \times \left( {8 - 1} \right) \times \left( {8 - 2} \right) \times .... \times 2 \times 1$
   Now simplifying the above expression, we get
   $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320$
   So $8!$ is equal to 40320.

2. Now we have to find the difference between two factorials.
   So, we will first find out the factorial of both the numbers and then apply the given mathematical operation.
   We will substitute ${\rm{n}} = 4$ in the formula ${\rm{n}}! = {\rm{n}} \times \left( {{\rm{n}} - 1} \right) \times \left( {{\rm{n}} - 2} \right) \times .... \times 2 \times 1$ to find $4!$.
   $\begin{array}{l}4! = 4 \times \left( {4 - 1} \right) \times \left( {4 - 2} \right) \times 1\\\ = 4 \times 3 \times 2 \times 1\\\ = 24\end{array}$
   Now, we will substitute ${\rm{n}} = 4$ in the formula ${\rm{n}}! = {\rm{n}} \times \left( {{\rm{n}} - 1} \right) \times \left( {{\rm{n}} - 2} \right) \times .... \times 2 \times 1$ to find $3!$.
   $\begin{array}{l}3! = 3 \times \left( {3 - 1} \right) \times 1\\\ = 3 \times 2 \times 1\\\ = 6\end{array}$
   We will substitute the values in the expression
   $4! - 3! = 24 - 6 = 18$

Note:
Note: We know that the factorial of a natural number ${\rm{n}}$ is denoted by ${\rm{n}}!$. It is important for us to remember that factorial is used to multiply numbers which are equal or less than that. It will help us evaluate the factorial even if we forget the formula of factorial of a number. Factorial operation is widely used in different areas of mathematics such as permutation and combination, algebra etc. It is also important to know that the factorial of 0 is 1.