Question

Evaluate \[5!4!\] = ?

Answer 1

Hint: We know that, ‘!’ sign used in maths means factorial. A factorial is a function that multiplies a number by every number below it. For solving this, we will use \[n! = n \times \left( {n - 1} \right)!\] where \[n \geqslant 1\] . So, we will evaluate first \[5!\] and then \[4!\] . After solving this, we will simplify this, and multiply both the factorial answers to get the final output.

Complete step-by-step answer:
Given that, \[5!4!\]
Factorials are just products. It is a multiplication operation of natural numbers with all the natural numbers that are less than it. It is denoted by \[n!\] where n is all positive integers.
The formula to find the factorial of a number is:
\[n! = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times \left( {n - 3} \right) \times \ldots . \times 3 \times 2 \times 1\]
Thus, the expansion of the formula gives the numbers to be multiplied together to get the factorial of the number.
Now, we will first evaluate only 5! As below:
\[\therefore 5!\]
We will apply \[n! = n \times \left( {n - 1} \right)!\] and we will get,
\[ = 5 \times (5 - 1)!\]
\[ = 5 \times 4!\]
\[ = 5 \times 4 \times (4 - 1)!\]
On simplifying this, we will get,
\[ = 5 \times 4 \times 3!\]
\[ = 5 \times 4 \times 3 \times (3 - 1)!\]
Simplify and remove the brackets, we will get,
\[ = 5 \times 4 \times 3 \times 2!\]
\[ = 5 \times 4 \times 3 \times 2 \times (2 - 1)!\;\]
\[ = 5 \times 4 \times 3 \times 2 \times 1!\]
We know that, \[1! = 1\] and so using this, we will get,
\[ = 5 \times 4 \times 3 \times 2 \times 1\]
\[ = 120\]
Now we will evaluate the second factorial number given 4! As below:
\[\therefore 4!\]
We know that, \[n! = 1 \times 2 \times 3 \times .... \times n\] and so will apply this, we will get,
\[ = 1 \times 2 \times 3 \times 4\]
\[ = 24\]
Thus,
\[\therefore 5!4!\]
\[ = 120 \times 24\]
\[ = 2880\]
Hence, the answer of \[5!4! = 2880\] .
So, the correct answer is “2880”.

Note: The factorial of a positive integer is represented by the symbol \[n!\]. Notice that \[0!\] is defined as 1 by mathematicians as it is the empty product. Thus, in short we can say that, the recurrence relation for the factorial of a number is defined as the product of the factorial number and factorial of that number minus 1.