Question

How do you simplify $72 \times 7!$ ?

Answer 1

Hint: Whenever they ask us to simplify the expression with some factorial number we can make use of the factorial expansion formula to simplify the factorial number and then performing the required operation we arrive at the correct answer. The factorial number expansion for $n$ is given by $n! = n \times (n - 1) \times (n - 2) \times (n - 3) \times ....... \times 1$ , here in this question substitute $7$ in place of $n$ to get the answer.

Complete step by step answer:
Whenever they ask us to simplify the expression with some factorial number we can make use of the factorial expansion formula to simplify the factorial number. Factorial of a number is nothing but the product of all positive integers less than or equal to a given number.
To find the factorial of a number we have a formula given by: $n! = n \times (n - 1) \times (n - 2) \times (n - 3) \times ....... \times 1$ .
Now to simplify $72 \times 7!$ first we solve for $7!$ then we multiply the answer with $72$.
Therefore, by using factorial formula we have
$7! = 7 \times (7 - 1) \times (7 - 2) \times (7 - 3) \times (7 - 4) \times (7 - 5) \times (7 - 6)$
$ \Rightarrow 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
On simplifying the above expression, we get
$ \Rightarrow 7! = 5,040$
Now to find the value of $72 \times 7!$ , we have
$72 \times 7! = 72 \times 5,040$
$ \Rightarrow 72 \times 7! = 3,62,880$
$3,62,880$ is nothing but the $9!$ . (if you do the factorial of $9$ you get the same answer)

Hence the simplified form of $72 \times 7!$ is $9!$.

Note:
The given problem can be solved in another way also. First, we find the factors of the number $72$ which are $9$ and $8$. If we multiply these two numbers we get $72$. Therefore, if we combine the factors $9$ and $8$ with $7!$, we directly get $9!$. So if you know the concept of factorial clearly then you can easily solve this type of problem.