Quick questions on Permutations: counting ordered arrangements with the multiplication principle

The number of ways to arrange all $n$ distinct objects in a row is

The number of ways to choose and arrange $r$ objects from $n$ distinct objects (order matters, no repetition) is

If you have $n$ objects of which $n_1$ are alike, $n_2$ are alike, $\dots$, $n_k$ are alike (with $n_1 + n_2 + \dots + n_k = n$), the number of distinct arrangements is

The number of distinct circular arrangements of $n$ distinct objects is

Glue them into a single block and arrange the block among the other objects: if there were $n$ objects, you now arrange $n - 1$ items, giving $(n - 1)!$, then multiply by the internal arrangements of the block ($2!$ for two glued objects, $m!$ for a block of $m$). "All the vowels together" is the same move: glue every vowel into one block.

Do not try to count this directly. Use the complement: (not together) $=$ (total) $-$ (together). This is almost always faster and is what markers expect.

Lock that object in place and arrange the rest. With one object fixed, you arrange the remaining $n - 1$ in $(n - 1)!$ ways.

Gluing two objects into a block fixes their position relative to the rest, but you must still multiply by $2!$ for the two orders inside the block (or $m!$ for a block of $m$).