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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