Fear factor!!

The prime factorization of $10!$ is $2^8\times3^4\times5^2\times7.$ We have $9$ ways to choose a power of $2$ for a factor ( $8+1$ ), $5$ ways to choose a power of $3$ ( $4+1$ ), $3$ ways for $5$ ( $2+1$ ), and $2$ ways for 7 ( $1+1$ ). Thus, the number of factors of $10!$ is $9\times5\times3\times2=\boxed{270}.$

Extra: This way of finding the number of factors of a number is known as the Formula of Factors.

Read more here: Number of Factors of an Integer

Let's factorize 10! = $1*2*3*4*5*6*7*8*9*10=$

$2^8*3^4*5^2*7$

To get the number of factors we take all exponents and add one to them, then multiply them.

$9*5*3*2 = \boxed{270}.$