What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$?

 I think you're saying

C(20, n) + C(20,10)  = C(21, 11)    if so....subtract C(20,10) fom both sides

C(20,n)  = C(21, 11) - C(20, 10)

C(20, n)  = 167960

And  ... C(20, 9)  = 167960    and therefore..... C(20, 11)   equals the same

So...the sum of these "n's"  =  9 + 11   = 20

I think that's correct, Mellie

Yup! It's correct. Thanks CPhill!!