This page illustrates how some categories of articles can be categorized in larger categories by means of combinatorics, binomial expansion, and Pascal's triangle. The section "One factor" is included for continuation of the pattern.

The pattern can be complicated by intermediate categories with names having phrases such as "by nationality" (or "by country"), "by religious status", "by sex", "by occupation", "by employment status", and "by marital status".

• Category:American scientists ⊆ Category:Scientists by nationality ⊆ Category:Scientists
• Category:American scientists ⊆ Category:American people by occupation ⊆ Category:Americans.

The guidelines at Wikipedia:Category names might not apply universal consistency to the order of elements in category names.

• Combination (1) of 1 element

X: Category:Scientists

• Combination (1) of 0 element

_: Category:Contents

• 1 + 1 = 2

• Combination (1) of 2 elements

XX: Category:American scientists

• Combinations (2) of 1 element

X_: Category:Americans

_X: Category:Scientists

• Combination (1) of 0 element

__: Category:Contents

• 1 + 2 + 1 = 4 = 2 × 2

• Combination (1) of 3 elements

XXX: Category:American women scientists

• Combinations (3) of 2 elements

XX_: Category:American women

X_X: Category:American scientists

_XX: Category:Women scientists

• Combinations (3) of 1 element

X__: Category:Americans

_X_: Category:Women

__X: Category:Scientists

• Combination (1) of 0 element

___: Category:Contents

• 1 + 3 + 3 + 1 = 8 = 2 × 2 × 2

• Combination (1) of 4 elements

XXXX: Category:American Jewish women scientists

• Combinations (4) of 3 elements

XXX_: Category:American Jewish women

XX_X: Category:American Jewish scientists

X_XX: Category:American women scientists

_XXX: Category:Jewish women scientists

• Combinations (6) of 2 elements

XX__: Category:American Jews

X_X_: Category:American women

X__X: Category:American scientists

_XX_: Category:Jewish women

_X_X: Category:Jewish scientists

__XX: Category:Women scientists

• Combinations (4) of 1 element

X___: Category:Americans

_X__: Category:Jews

__X_: Category:Women

___X: Category:Scientists

• Combination (1) of 0 element

____: Category:Contents

• 1 + 4 + 6 + 4 + 1 = 16 = 2 × 2 × 2 × 2

• Combination (1) of 5 elements

XXXXX: Category:Lists of American Jewish women scientists

• Combinations (5) of 4 elements

XXXX_: Category:Lists of American Jewish women

XXX_X: Category:Lists of American Jewish scientists

XX_XX: Category:Lists of American women scientists

X_XXX: Category:Lists of Jewish women scientists

_XXXX: Category:American Jewish women scientists

• Combinations (10) of 3 elements

XXX__: Category:Lists of American Jews

XX_X_: Category:Lists of American women

XX__X: Category:Lists of American scientists

X_XX_: Category:Lists of Jewish women

X_X_X: Category:Lists of Jewish scientists

X__XX: Category:Lists of women scientists

_XXX_: Category:American Jewish women

_XX_X: Category:American Jewish scientists

_X_XX: Category:American women scientists

__XXX: Category:Jewish women scientists

• Combinations (10) of 2 elements

XX___: Category:Lists of Americans

X_X__: Category:Lists of Jews

X__X_: Category:Lists of women

X___X: Category:Lists of scientists

_XX__: Category:American Jews

_X_X_: Category:American women

_X__X: Category:American scientists

__XX_: Category:Jewish women

__X_X: Category:Jewish scientists

___XX: Category:Women scientists

• Combinations (5) of 1 element

X____: Category:Lists

_X___: Category:Americans

__X__: Category:Jews

___X_: Category:Women

____X: Category:Scientists

• Combination (1) of 0 element

_____: Category:Contents

• 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2 × 2 × 2 × 2 × 2

• Combination (1) of 6 elements

XXXXXX: Category:Lists of award-winning American Jewish women scientists

• Combinations (6) of 5 elements

XXXXX_: Category:Lists of award-winning American Jewish women

XXXX_X: Category:Lists of award-winning American Jewish scientists

XXX_XX: Category:Lists of award-winning American women scientists

XX_XXX: Category:Lists of award-winning Jewish women scientists

X_XXXX: Category:Lists of American Jewish women scientists

_XXXXX: Category:Award-winning American Jewish women scientists

• Combinations (15) of 4 elements

XXXX__: Category:Lists of award-winning American Jews

XXX_X_: Category:Lists of award-winning American women

XXX__X: Category:Lists of award-winning American scientists

XX_XX_: Category:Lists of award-winning Jewish women

XX_X_X: Category:Lists of award-winning Jewish scientists

XX__XX: Category:Lists of award-winning women scientists

X_XXX_: Category:Lists of American Jewish women

X_XX_X: Category:Lists of American Jewish scientists

X_X_XX: Category:Lists of American women scientists

X__XXX: Category:Lists of Jewish women scientists

_XXXX_: Category:Award-winning American Jewish women

_XXX_X: Category:Award-winning American Jewish scientists

_XX_XX: Category:Award-winning Americanh women scientists

_X_XXX: Category:Award-winning Jewish women scientists

__XXXX: Category:American Jewish women scientists

• Combinations (20) of 3 elements

XXX___: Category:Lists of award-winning Americans

XX_X__: Category:Lists of award-winning Jews

XX__X_: Category:Lists of award-winning women

XX___X: Category:Lists of award-winning scientists

X_XX__: Category:Lists of American Jews

X_X_X_: Category:Lists of American women

X_X__X: Category:Lists of American scientists

X__XX_: Category:Lists of Jewish women

X__X_X: Category:Lists of Jewish scientists

X___XX: Category:Lists of women scientists

_XXX__: Category:Award-winning American Jewish

_XX_X_: Category:Award-winning American women

_XX__X: Category:Award-winning American scientists

_X_XX_: Category:Award-winning Jewish women

_X_X_X: Category:Award-winning Jewish scientists

_X__XX: Category:Award-winning women scientists

__XXX_: Category:American Jewish women

__XX_X: Category:American Jewish scientists

__X_XX: Category:American women scientists

___XXX: Category:Jewish women scientists

• Combinations (15) of 2 elements

XX____: Category:Lists of award-winners

X_X___: Category:Lists Americans

X__X__: Category:Lists of Jews

X___X_: Category:Lists of women

X____X: Category:Lists of scientists

_XX___: Category:Award-winning Americans

_X_X__: Category:Award-winning Jews

_X__X_: Category:Award-winning women

_X___X: Category:Award-winning scientists

__XX__: Category:American Jews

__X_X_: Category:American women

__X__X: Category:American scientists

___XX_: Category:Jewish women

___X_X: Category:Jewish scientists

____XX: Category:Women scientists

• Combination (6) of 1 element

X_____: Category:Lists

_X____: Category:Award-winners

__X___: Category:Americans

___X__: Category:Jews

____X_: Category:Women

_____X: Category:Scientists

• Combination (1) of 0 element

______: Category:Contents

• 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64 = 2 × 2 × 2 × 2 × 2 × 2

• Wikipedia:Category intersection