$X_1, \dots, X_n$ iis geometric: $P(X=x) = (1-p)^{x}p$, $x=0,1,2, \dots$
My Attempt:
$T=\sum_{i=1}^n X_i$ is a sufficient statistic
$W= \begin{cases}1 & X_1= 0,\\ 0 & X_1\neq 0\end{cases}$ W is an unbiased estimator of $p$
To find UMVUE, \begin{align} E[W|T=t] &= \frac{P(X_1 = 0, T=t)}{P(T=t)}\\[5pt] &= \frac{P(X_1 = 0)P(X_1+\cdots +X_n=t)}{P(T=t)}\\[5pt] \end{align} Can somebody please help me expand this step. It's confusing whether I should use $t-1$ or $t-2$ in the combination part of negative binomial pdf.