+4 What is the sum of all integer values \(n\) for which\( \binom{26}{13}+\binom{26}{n}=\binom{27}{14}\)?
+23245 26! / (13! · 13! ) + 26! / ( n! · (26 - n)! ) = 27! / ( 14! · 13! )
Rewrite as: 26! / (13! · 13! ) + 26! / (n! · (26 - n)! ) = 27 · 26! / ( 14 · 13! · 13! )
Divide by 26!: 1 / (13! · 13! ) + 1 / (n! · (26 - n)! ) = 27 / ( 14 · 13! · 13! )
1 / ( n! · (26 - n)! ) = 27 / ( 14 · 13! · 13! ) - 1 / (13! · 13! )
Factor: = 1 / (13! · 13! ) · ( 27/14 - 1 )
Simplify: = 1 / (13! · 13! ) · ( 13/14 )
= 13 / ( 14! · 13! )
= 1 / ( 14! · 12! )
Therefore, n! · (26 - n)! = 14! · 12!
So, n is either 12 or 14 ...
