What is \[(-4)!\,?\]

Hint: We are given a question based on factorial. We can see that the given number has a negative to it, that is, we have to find the factorial of a negative number. We know that factorial can be found only for positive integers and not for negative integers. We can back our statement with the definition of the factorial of a number. That is to say that, factorial of the given negative number does not exist.

Complete step by step answer:
According to the given question, we are given a question based on factorial. From the question itself, we can see that in the question the number has a negative sign. That is, we have to find the factorial of a negative number.
The given number we have is,
\[(-4)!\]
Factorial can be said to be a function which multiplies a number by every number below it.
For example – if we want to find the factorial of 5, that is, \[5!\]
We will get the factorial as, \[5!=5\times 4\times 3\times 2\times 1\]
For every number, we multiply the number below it up till 1 only. And if we go beyond 1, that is, 0 and other negative numbers, the result will be 0 only.
By definition of a factorial itself, which is, the factorial of a non – negative integer ‘n’ is represented as \[n!\].
On expanding \[n!\], we will have,
\[n!=n\times (n-1)\times (n-2)\times ...\times 1\]
so, we are implying that a factorial of a negative number cannot be found.

Therefore, the factorial of \[(-4)!\] does not exist.

Note: The question should be read carefully taking note of the sign on the number given. If we had missed the sign on the number given and had straight away gone to apply the factorial on the number, we would have got an incorrect answer.