= Solution
The estimator $U=X_1X_2$ is <unbiased estimator>[unbiased] for $p^2$ because the <independent random variables> $X_1,X_2$ satisfy $\mathbb E[X_1X_2]=p^2$. Given $T=t$, all placements of the $t$ successes are equally likely, so
$$
\mathbb E[X_1X_2\mid T=t]
=\frac{\binom{n-2}{t-2}}{\binom nt}
=\frac{t(t-1)}{n(n-1)}.
$$
Thus the Rao-Blackwellized estimator is
$$
\boxed{\widehat{p^2}=\frac{T(T-1)}{n(n-1)}}.
$$
It is unbiased by the tower property. Since $n\geq3$ and $p\in(0,1)$, the event $T=2$ has positive probability, and conditionally on it $X_1X_2$ takes both zero and one with positive probability. Hence $\mathbb E[\operatorname{Var}(X_1X_2\mid T)]>0$, so the new estimator has strictly smaller variance.
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