I have the following question. I have a set of elements $A,B,C$ like this: $$ A_nB_mC_l $$ $n,m,l$ are the element counts.
How to calculate all possible combinations (without replacement, the order is not important) according to $n,m,l$?
For example if I have $A_2B_2C$ there are following combinations possible:
amount of elements = 1: $A,B,C$ - 3 combinations
amount of elements = 2: $A_2,B_2,AB,AC,BC$ - 5 combinations
amount of elements = 3: $A_2B,A_2C,AB_2,B_2C,ABC$ - 5 combinations
amount of elements = 4: $A_2BC,AB_2C,A_2B_2$ - 3 combinations
amount of elements = 5: $A_2B_2C$ - 1 combination
So, in total I have $3+5+5+3+1=17$ combinations