2016 is absolutely awesome 16

Algebra Level 5

\[x^{2016}+cx^{2015}+\binom{c}{2}x^{2014}+\binom{c}{3}x^{2013}+\cdots+\binom{c}{2016}\]

How many real numbers \(c\) are there such that the above polynomial has 2016 real roots (counting multiplicities)?

Details and Assumptions

  • We define the generalized binomial coefficient as \(\displaystyle\binom{c}{k}=\dfrac{c(c-1)(c-2)\cdots(c-k+1)}{k!}\).

  • If there are infinite real numbers \(c\) satisfied, submit \(-1\) as the answer.


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