The product rule counts arrangements as if every item were different, which is how permutations work. When some arrangements look the same (repeated letters, seats around a round table), the product rule overcounts. If the overcount is the same for every object, one division fixes it.
Key idea
Division rule: if a method produces every object exactly $d$ times, the number of objects is (the method's count) $\div\ d$. Arrangements of $n$ letters with one letter repeated $k_{1}$ times, another $k_{2}$ times, and so on: $\dfrac{n!}{k_{1}!\,k_{2}!\cdots}$. Seatings of $n$ people around a round table (rotations counted as the same): $(n - 1)!$.
Why it works
Give the copies of each letter temporary labels, like $A_{1}A_{2}A_{3}$ in BANANA. Now all letters are different, and there are $6! = 720$ labeled arrangements. Erasing the labels merges arrangements that differ only in how the copies are labeled: $3!$ orders of the A's times $2!$ orders of the N's, the same $12$ for every word. So there are $720 / 12 = 60$ words. For a round table, each seating appears $n$ times among the $n!$ row orders, once for each place where you could cut the circle.
Worked examples
Worked example: A long word
How many arrangements of MISSISSIPPI are there?
- $11$ letters: M once, I four times, S four times, P twice.
- Labeled arrangements: $11!$. Each word appears $4! \cdot 4! \cdot 2!$ times.
- $\dfrac{11!}{4!\,4!\,2!} = \dfrac{39{,}916{,}800}{24 \cdot 24 \cdot 2} = 34{,}650$.
Answer: $34{,}650$
Worked example: A round table
In how many ways can $6$ people sit around a round table, if seatings that are rotations of each other count as the same?
- In a row: $6! = 720$ orders.
- Each round seating is counted $6$ times, once for each rotation.
- $720 / 6 = 120 = 5!$. Equivalently: seat one person anywhere, then arrange the other $5$ relative to them.
Answer: $120$
Worked example: Rotations and flips
How many different bracelets can be made from $5$ different beads, if a bracelet can be rotated or flipped over?
- Around a circle, up to rotation: $(5 - 1)! = 24$.
- Flipping turns each arrangement into its mirror image. For $5$ different beads, an arrangement never matches its own mirror image, so each bracelet was counted twice.
- $24 / 2 = 12$.
Answer: $12$
Common mistake
Dividing by the wrong factorials, or not dividing for rotations. For BANANA, dividing by $3!$ alone (for the A's) forgets the N's; dividing by $6!/2$ or by $2!$ for each letter makes no sense. And at a round table, $6!$ counts rotations as different seatings. Check: divide by one factorial for each repeated letter, using that letter's count; and ask whether two outcomes your method separates would look identical.