when would you multiply permutations?

hey all,

so, I was practicing some permutation problems and this one came up.

'A company is trying to fill some vacancies. four of five accountants will be selected and two of four lawyers will be selected. how many different ways can they be selected?'

I found out both permutations, but then I was stuck on what to do. The answer key said I should multiply them because the different permutations are independent of one another. Is this true? if so, why? and what happens if the permutations are dependent?

thanks for your help!

Re: when would you multiply permutations?

Hey hellothisismyname.

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).

Re: when would you multiply permutations?

Thanks, that really cleared it up for me. If the two variables are dependent, is their any way you can use permutations/combinations in a similar manner? or is it simply not feasible to do so?

Re: when would you multiply permutations?

Not really unless you have more information.

You need enough information about the association and if you don't have it you either need to introduce assumptions yourself that make the most sense or get enough data to create a model that makes sense.