Anagram Counting

INFO

An anagram is a rearrangement of a multiset of characters.
When characters repeat, we divide by the factorial of their frequencies to avoid overcounting identical arrangements.

---

General Formula

If a word has $n$ total letters, and some letters repeat with frequencies $f_1,f_2,\dots ,f_k$, then the number of distinct anagrams is:

$$\frac{n!}{f_1!\cdot f_2!\cdot \cdots \cdot f_k!}$$

---

Examples

1. Letters {E, E, T, S}

• Total permutations of labeled characters: $4!$
• Repetition of E counted $2!$ times

$$\frac{4!}{2!}=12$$

2. Letters {O, O, F, F, N}

• Total permutations: $5!$
• Two letters (O and F) each repeated twice

$$\frac{5!}{2!\cdot 2!}=30$$

3. Word: REASSESS

• Letters: R, E, A, S, S, E, S, S
• Frequencies: R (1), E (2), A (1), S (4)

$$\frac{8!}{1!\cdot 1!\cdot 2!\cdot 4!}=8\cdot 7\cdot 5\cdot 3=840$$

NOTE

This is a classic use of the QuotientRule.