The coefficients \(C_{k}^{p}\) in the binomial expansion for \((1+x)^{p}\) are given by

\[C_{k}^{p}=\frac{p(p-1) \cdots(p-k+1)}{k!}\]

a. Write \(C_{k}^{p}\) in terms of Gamma functions.

b. For \(p=1 / 2\), use the properties of Gamma functions to write \(C_{k}^{1 / 2}\) in terms of factorials.

c. Confirm your answer in part

b. by deriving the Maclaurin series expansion of \((1+x)^{1 / 2}\).