Conditional distributions of binomial sums

Let $X_1,X_2,X_3$ be three independent and identically distributed binomial random variables with number of trials $n = 100$ and success probability $p$ ($0 < p < 1$), which is an unknown parameter. Let $T_1 = (X_1 + X_2, X_3)$ and $T_2 = X_1 + X_2 + X_3$. Consider the following statements: (I) The distribution of $T_2$ given $T_1 = t_1$ is independent of $p$. (II) The distribution of $T_1$ given $T_2 = t_2$ is independent of $p$. Which of the above statements is/are true?