Total Possible Combinations of Combinations

[gurs]

I am trying to calculate the total number of possible combinations of a set of variables. The calculation is complicated by the fact that not every variable behaves the same way.

Here are the details. I have 5 different trait types. Each trait type has a different number of possible traits. Some traits can occur only once per observation, some can occur once or twice, and some might not occur at all. Here is an example of the data set:

• Trait1 (hair color): 3 possible states; 1 state MUST be present in all observations
• Trait2 (talent type): 3 possible states; 1 or 2 states MUST be present in all observations
• Trait3 (location): 2 possible states; 1 state MUST be present in all observations
• Trait4 (disability): 3 possible states; 1 state MAY be present in all observations (so Trait4 may be absent)
• Trait5 (marital status): 2 possible states; 1 state MUST be present in all observations

How would I calculate the total number of possible combinations of traits 1 through 5? In truth, I think this is as much a math question as an Excel question!

 

[JGordon11]

3*(3*3)*2*4*2 = 432

 

[gurs]

3*(3*3)*2*4*2 = 432

Got it. So I basically treat Trait2 as two different traits, and treat Trait4 as having 4 possible states (3 actual states plus a null state). Is that your math?

 

[JGordon11]

Yes, that's correct.

 

[gurs]

Thanks!

 

[gurs]

Aren't we potentially overcounting if we consider Trait2 to be two different traits? Your math is to count Trait2 as 3x3=9, but there is no state in which a subject is counted as having both 1 state AND 2 states. They have either 1 OR 2. So there should only be 6 possible combinations for Trait2 (3 possible states; 1 or 2 states must be present in all observations):

TALENT TYPE COMBOS (Types=A,B,C):​

A​

B​

C​

AB​

AC​

BC​

What is the formula for this result if the number of types and potential stacking is different? For example, what about a trait with 4 possible states, in which 2, 3 or 4 states must be present in all observations. What is the formula to calculate those combinations? While the example I used in this post is simple, my use-case is much more complicated, so I need a way to structure the inputs and formula in a repeatable fashion.

 

[JGordon11]

I agree it should be (3+3) not (3*3). If it were 4 types with either 1 or 2 it would be (4+6). In general if it were n types with either a1, a2, ... aj it would be
(Combin(n,a1) + Combin(n,a2) + ... + Combin(n,aj))

 

[gurs]

Thanks for that. So the total number of combinations in my initial example is 3 * (3 * 2) * 2 * 4 * 2 = 288. Yes?

 

[JGordon11]

?