What is the distribution of the partition sizes of a randomly partitioned tuple?

Question

Three equivalent formulations of my problem:

1. Suppose we have a tuple of size $n$ and partition it into $m$ parts so that the $m-1$ partition points are uniformly distributed over all possible partitions. What is the distribution of the cardinality of any of the partitions?

2. Suppose we are given $m$ identically independently distributed (by condition positive) Poisson random variables $X_i\in\mathbb N_+$ given $\sum_i^m X_i = n$. What is the distribution of $X_i$?

3. Suppose we toss a coin $n$ times and are given that Heads occurs exactly $m-1$ times. What is the distribution of the number of coin tosses until the first head is tossed?

I know that the joint distribution of the $X_i$ follows a multinomial distribution with parameters $n$ and $p_i=\frac{1}{m}$. Consequently,

$$\Pr(X_i = k)= \sum_{k+k_2+\dots+k_m=n,\; k_i\geq 1}\frac{1}{m^n}{{n}\choose{k,k_2,\dots,k_{m}}} $$

However, as I am interested in the distribution of only one $X_i$, I am wondering if there is a nice closed form solution for this nasty sum?

Edit

I am not sure if the multinomial distribution is actually appropriate, because the compartments follow a strict ordering. That is, an element at the end of the tuple is less likely to be in the first compartment than in the last.