I have 5 categories - A, B, C, D & E and each of these categories have sub categories like following

A - { a, b}

B - { c, d, e}

C - { f, g }

D - { h , i}

E - { j, k ,l}

I want to find the total combinations(ones+doubles+trebles+fours+fives) of the above system without repetitions

In each combinations, selecting elements from same sub category is not allowed like {Aa, Ab} cannot be in one combination

For Example
Combination of one's will be like below

Aa

Ab

Bc

....so on we have 12 such combinations for ones

Combination of two's

Aa , Bc

Ab , Bc,

Cf , Di

..... So on we will have some number of combinations
Here we cannot have {Aa,Ab} in one of the combinations. {Aa,Ab} is not a valid combination

Combination of three's

Aa, Bc, Cf

Bc, Dh, Ej

.....So on 
Here we cannot have {Aa,Ab,Bc} in one of the combinations. {Aa,Ab,Bc} is not a valid combination , you must choose from different categories

Combination of fours'

Aa,Bc,Cf,Dh

Aa,Bc,Cf,Di

Aa,Bc,Cg,Dh

.......So on
Here we cannot have {Aa,Ab,Bc,Cf} in one of the combinations. {Aa,Ab,Bc, Cf} is not a valid combination , you must choose from different categories

Combination of five's

Aa,Bc,Cf,Dh,Ej

Aa,Bc,Cf,Dh,El

..... So On 
Here also elements from same category should not be in a combination

I want to find the total number of combinations i.e 
Total = No of Combinations of { one's + two's + three's + four's + five's}

Similarly if I have large number of categories like 15 instead of 5, Any generalized formula to calculate the number of combinations?