= Solution
A <sufficient statistic> $T$ is one for which the conditional distribution of the full sample $(X_1,\ldots,X_n)$ given $T$ does not depend on $p$. Take
$$
T=\sum_{i=1}^nX_i.
$$
For a binary sample $x$ with $\sum_i x_i=t$, its <likelihood function> is
$$
p^t(1-p)^{n-t},
$$
which depends on the data only through $t$. By the <Fisher-Neyman factorization theorem>, $T$ is sufficient. Equivalently, conditionally on $T=t$, the sample is uniform over the $\binom nt$ binary vectors containing $t$ ones, independently of $p$.
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