Words, repeats and ranks

Build three tools for word puzzles, and check them against brute force.

The idea. A word with repeated letters has $\dfrac{n!}{p!\, q!\cdots}$ arrangements. The rank of a word with different letters counts the words before it, one place at a time.

Your tasks.

1. In arrangements, return the number of different arrangements of the letters of word. Use Counter and math.factorial.
2. In together, return the number of rows of n different people in which k given people stand side by side.
3. In rank, return the place of word in dictionary order, for a word whose letters are all different.