Quotient Rule

INFO

If a set $A$ can be partitioned into disjoint subsets $X_1,X_2,\dots ,X_k$ each having the same number of elements,
that is, $|X_1|=|X_2|=\cdots =|X_k|$,
then the number of subsets is:

$$k=\frac{|A|}{|X_1|}$$

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When to Use the Quotient Rule

This rule is useful when a set can be expanded into equally sized subsets, and each subset is easy to count. It helps eliminate overcounting due to symmetry or repeated structure.

Common applications include:

• Anagram counting: dividing total permutations by repeated character counts
  → See Anagram Counting
• Counting objects with orientation: e.g., necklaces, bracelets, or rotational symmetries
• Combinations of objects: when multiple arrangements map to the same outcome
• Fixed-density binary strings: e.g., strings with exactly $k$ ones and $n-k$ zeros

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Example: Anagram

How many distinct anagrams of the word REASSESS?

• Total letters: 8
• Frequencies: R (1), E (2), A (1), S (4)

$$\frac{8!}{1!\cdot 2!\cdot 1!\cdot 4!}$$

NOTE

The denominator accounts for indistinguishable permutations due to repeated letters.