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What is the sum of all integer values \(n\) for which \(\binom{26}{13}+\binom{26}{n}=\binom{27}{14}\)?

 Dec 8, 2018
 #1
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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)

 

So

 

n = 12    or n = 14

 

And their sum =  26

 

 

cool cool cool

 Dec 8, 2018

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