What is the sum of all integer values \(n\) for which \(\binom{26}{13}+\binom{26}{n}=\binom{27}{14}\)?
Using Pascal's Triangle
C( n , k) + C(n, k + 1) = C(n+1, k + 1)
So
C(26, 13) + C(26, 14) = C( 27, 14)
But ....C(27, 14) = C(27, 13) ....so we also have
C(26, 12) + C(26, 13) = C(27, 13)
n = 12 or n = 14
And their sum = 26
+1206 
