How many factors does 6! have?

There are two ways to solve this problem:

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$1)$ $6!=6*5*4*3*2*1$ , and we have to prime factorize this.

This means that it is for $6$ , $3*2$ .

For $5$ , $5$ .

For $4$ , $2*2$ .

For $3$ , $3$ .

And for $2$ , $2$ .

So the prime factorization of $6!$ is $3*2*5*2*2*3*2$ , and this is $2^4*3^2*5^1$ .

And if we add 1 to each of the exponents, in this case 4, 2, and 1, we get 5, 3, and 2.

If we multiply all of these numbers, $5*3*2$ , we get $\boxed{30}$ .

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$2)$ We could calculate $6!$ and get that it is $720$ , and do the same process as we did above.