# Binomially Amazing!

Calculus Level 5

$(1+x)^{1729} = C_0 + C_1 x +C_2 x^2 + \ldots + C_{1729} x^{1729}$

The equation above is an algebraic identity for constants $$C_0,C_1,\ldots,C_{1729}$$. Given that

$\frac{C_0}1 - \frac{C_1}5 + \frac{C_2}9 - \ldots - \frac{C_{1729}}{6917}$

is equal to $$\dfrac{a! \ 4^b}{c!!!!}$$, where $$a,b$$ and $$c$$ are integers, find the value of $$a+b+c$$.

Clarification:

• $$n!!!!$$ denotes the quadruple factorial, eg $$15!!!! = 15\times11\times7 \times3$$.

• $$1,5,9,\ldots,6917$$ follows an arithmetic progression.

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