Its easier to show this in terms of probability. We can relate probabilities to frequencies by noting that a probability of something P(A) = #Frequency of A occuring / #Total Number of Things.
In probability we have a theorem that says if A and B are independent then P(A and B) = P(A)P(B).
The proof of this is based on conditional probability. We define the probability of A given B as P(A|B) = P(A and B)/P(B).
Lets consider two random events A and B. If they are independent, then it means that knowing one of them won't affect the probability of the other one. In other words lets say you had the probability of selecting a lawyer and the probability of selecting an accountant: in this case the two events would be independent if knowing one result didn't have any effect on the other.
Mathematically we write this as P(A|B) = P(A) and P(B|A) = P(B). In other words, any extra information about anything outside of what you are looking at doesn't have any impact on the probability which means that there is no causal effect between two things.
If things were not independent then changing one thing would affect the other thing. But if they are independent then changing one thing doesn't change the other. That is the intuitive idea behind independence.
So P(A|B) = P(A and B)/P(B), if P(A|B) = P(A) then re-arranging gives us:
P(A|B) = P(A) = P(A and B)/P(B) so this implies P(A and B) = P(A)P(B) if P(B) != 0.
Since probabilities are relative frequencies, you can use this result for permutations and combinations if you are dealing with independent things.
If for example you have a total number of possibilities as N then you can convert a probability to a frequency by multiplying it by N. In other words:
Freq_A = P(A)*N.
The above should give you an idea of why it works (i.e. the multiplication rule).