Simplified form for sum of permutations ? Combinitorics question

I have been beating my head against the wall, because I know there
must be a simplified generalized formula for what I want to do.

If I have a group of values, say a,b,c,d


I want to sum the product of all these permutations. basically:


ab+ac+ad+bc+bd+cd, which is 11 math operations


which I think is best simplified as (a+b)(c+d) + ab + cd (7 math
operations)


Or perhaps:


ab + (a+b)c + (a+b+c)d, which can also be thought of as 7 operations.


But, what is the general form when the I want to chose 3 out of my
four values (abc+abd+bcd), or groups of 10 out of 100 total, etc?


Thanks

.

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