Ghost Writer II

Fix positive integers $n, k,$ and $m$ with $1 < k < n$ and $m \geq k-1$ . Suppose an alphabet has $n$ distinct letters $a_1,\dots, a_n$ . A $\textit{word}$ is any non-empty string of characters in the alphabet, and a $\textit{sentence}$ of length $j$ is a collection of $j$ words separated by any amount of spaces that does not start nor end with a space. How many distinct sentences of length $k$ can be made by using every letter in the alphabet exactly once and exactly $m$ spaces? Two sentences are considered non-distinct if the following conditions are met: Both sentences have the same number of words, say $w$ . The $i$ th word in the first sentence is the same as the $i$ th word in the second sentence and has the same number of spaces trailing it for all $i$ . Report the answer when $k = 3, n = 6,$ and $m = 5$ .

Answer format: enter an integer.