More fun in 2016, Part 15
Find S = k = 0 ∑ 2 0 1 6 ( − 1 ) k ( k 2 0 1 6 ) ( k + 2 0 1 5 ) 2 0 1 6
Write S = Γ ( x ) for a real number x > 1 enter x .
The answer is 2017.
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1 solution
Exactly! (+1)
I did not even think about the red and blue boxes... I just observed that the term "...+2015" is immaterial, based on the previous problem
Here is a more challenging one
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Wait I'm intrigued. What do you mean by " is immaterial"?
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( k + 2 0 1 5 ) 2 0 1 6 = k 2 0 1 6 + a polynomial of lower degree. You showed in Part 14 that the corresponding sum for that polynomial will be 0 so k = 0 ∑ 2 0 1 6 ( − 1 ) k ( k 2 0 1 6 ) ( k + 2 0 1 5 ) 2 0 1 6 = k = 0 ∑ 2 0 1 6 ( − 1 ) k ( k 2 0 1 6 ) k 2 0 1 6 = 2 0 1 6 !
Just define a few terms of this I.E.(inclusion exclusion)...As I think the condition on number of boxes and being non empty is just flipped.
By the Principle of Inclusion and Exclusion (PIE) , this is equivalent to arranging 2016 distinct balls in a row of 2015 distinct red boxes and 2016 distinct blue boxes such that the blue boxes are not empty. Obviously, this is just 2016!