Pascal's Triangle and Combinatorics
Pascal's Triangle can be used to easily work out the number of permutations for a given number of "ingredients" and "places". For example the above diagram highlights that the number of permutations for 3 ingredients over 3 places equals 27:
AAA  AAB  ABA  BAA  AAC  ACA  CAA  ABC  ACB 
BBB  BBA  BAB  ABB  BBC  BCB  CBB  BCA  BAC 
CCC  CCA  CAC  ACC  CCB  CCB  BCC  CAB  CBA 
One simply follows the diagonals down from each side. The circle where they meet shows the answer. The smaller number in the circle indicates the number of different Ratios possible for the ingredients and places. In this case it is 10:
ABC  
AAA  BBB  CCC 
AAB  BBA  CCA 
AAC  BBC  CCB 
Note that the small black "Ratio" numbers are identical with the numbers of Pascal's Triangle.
The larger blue "Permutation" numbers follow a simple pattern. The diagonal line from a particular Ingredient Number contains all the powers of that number from 1 to as far as the triangle is extended. For example the 2 Ingredient Line contains all the powers of 2 ie: 1,2,4,8,16,32...etc and the 3 Ingredient Line contains all the powers of 3 ie: 1, 3, 9, 27, 81...etc.
The diagnal line from a particular Place Number contains all the numbers of a particular power from 1 to as far as the triangle is extended. For example the 2 Place Line includes all the Square Numbers ie: 1, 4, 9, 16, 25...etc and the 3 Place Line contains all the cube numbers ie: 1, 8, 27, 64...etc.
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