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This page illustrates how categories of articles can be categorized in larger categories by means of combinatorics, binomial expansion, and Pascal's triangle.

Two factors

  • Combination (1) of 2 factors
XX: Category:American scientists
  • Combinations (2) of 1 factor
X_: Category:Americans
_X: Category:Scientists
  • Combination (1) of 0 factor
__: Category:Contents
  • 1 + 2 + 1 = 4 = 2 × 2

Three factors

  • Combination (1) of 3 factors
XXX: Category:American women scientists
  • Combinations (3) of 2 factors
XX_: Category:American women
X_X: Category:American scientists
_XX: Category:Women scientists
  • Combinations (3) of 1 factor
X__: Category:Americans
_X_: Category:Women
__X: Category:Scientists
  • Combination (1) of 0 factor
___: Category:Contents
  • 1 + 3 + 3 + 1 = 8 = 2 × 2 × 2

Four factors

  • Combination (1) of 4 factors
XXXX: Category:American Jewish women scientists
  • Combinations (4) of 3 factors
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 factors
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 factor
X___: Category:Americans
_X__: Category:Jews
__X_: Category:Women
___X: Category:Scientists
  • Combination (1) of 0 factor
____: Category:Contents
  • 1 + 4 + 6 + 4 + 1 = 16 = 2 × 2 × 2 × 2

Five factors

  • Combination (1) of 5 factors
XXXXX: Category:Lists of American Jewish women scientists
  • Combinations (5) of 4 factors
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 factors
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 factors
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 factor
X____: Category:Lists
_X___: Category:Americans
__X__: Category:Jews
___X_: Category:Women
____X: Category:Scientists
  • Combination (1) of 0 factor
_____: Category:Contents
  • 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2 × 2 × 2 × 2 × 2

Six factors

  • Combination (1) of 6 factors
XXXXXX: Category:Lists of award-winning American Jewish women scientists
  • Combinations (6) of 5 factors
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 factors
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 factors
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 factors
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 factor
X_____: Category:Lists
_X____: Category:Award-winners
__X___: Category:Americans
___X__: Category:Jews
____X_: Category:Women
_____X: Category:Scientists
  • Combination (1) of 0 factor
______: Category:Contents
  • 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64 = 2 × 2 × 2 × 2 × 2 × 2
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