Medium Practice — Set 1

Quick Recap — Arrangements with Repetition & Restrictions

• Repeated letters: arrangements of $n$ objects with repeats $p,q,\dots$ number $\dfrac{n!}{p!\,q!\cdots}$.
• Circular: $n$ distinct objects around a table $=(n-1)!$; on a ring (reflections same) $=\tfrac{(n-1)!}{2}$.
• Together / not together: glue a group as one block (then multiply by internal arrangements), or use the gap method.
• With repetition allowed, $r$ positions from $n$ symbols $=n^r$.