How to determine the number of unique subsets of 2,3,5 in a set of 7 or 8?

Question

I am trying to determine the number of unique subsets of 2,3,5 in each case of a set of seven and a set of eight...

By unique subset I mean: In an example set of three (1,2,3) there would be three unique subsets of 1 (1),(2),(3) and three unique subsets of 2 (1,2),(1,3),(2,3) (simply inverting or rearranging the components would not be counted as unique, as in (1,2) (2,1)...

I know that I can do this manually because the numbers are small, but I would rather know the proper steps to follow to get a correct answer for those times I need unique subsets of seven from a set of six thousand...

Sub question: Does a set of three (1,2,3) contain a subset of three (1,2,3)???

Answer 1

SET OF SEVEN

C(7,5)=C(7,2) or 7C5 or 7C2=21 sets of 2 or 5 (=7*6/2=7*6*5*4*3/(1*2*3*4*5)).

C(7,3)=7C3=35 sets of 3 (=7*6*5/(1*2*3)).

SETS OF EIGHT

C(8,2)=28 sets of 2; C(8,3)=C(8,5)=56 sets of 3 or 5.

The palindromic pattern follows Pascal's Triangle for row 7 and row 8;

ROW 7: 1 7 21 35 35 21 7 1 = C(7,0) C(7,1) C(7,2),..., C(7,6) C(7,7), and the sum is 2^7=128

ROW 8: 1 8 28 56 70 56 28 8 1

I don't think { 1 2 3 } can be considered a subset of { 1 2 3 } but the subsets will include the null set { }; the set itself is included in the power set, and the total number of sets in the power set is 2^3=8 in this case and 2^n in the general case where n is the set size.