How many different positive integer factors does have? How many different positive integer factors does $(2^7)(3^4)(7^3)(23^5)$ have?
Do we have to do any combinations between the powers here? 

What you are looking for is the number of the positive divisors of $2^7\cdot 3^4\cdot 7^3\cdot 23^5$. Then, the answer is
$$(7+1)(4+1)(3+1)(5+1)=8\cdot 5\cdot 4\cdot 6=960.$$
In general, if
$$N={p_1}^{\color{red}{q_1}}\cdot {p_2}^{\color{red}{q_2}}\cdots {p_k}^{\color{red}{q_k}}$$
where $q_i,k\in\mathbb N$ and $p_1\lt p_2\lt\cdots\lt p_k$ are primes, then the number of the positive divisors of $N$ can be represented as
$$(\color{red}{q_1}+1)(\color{red}{q_2}+1)\cdots (\color{red}{q_k}+1).$$ 
