How many way I can peak up from an array

This is equivalent to star bar problem. I want to implement it in matlab.

I have an array A which has [1 1 1 1......1] 20 elements I have an array B which has [2 2 2 2......2] 20 elements I have an array C which has [3 3 3 3......3] 20 elements

Now I want to find in how many ways I can select 3 to 9 elements from all these array..

like i can take 3 from A or 3 from B or 3 from C or it can be 1 from A and 2 from B...like this...so on......

abc=[A B C]; 
allcombs = nchoosek(abc, 9);
combs  = unique(allcombs, 'rows');

I don't think its a correct approach...It will computationally tough.