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Combinatorics - Wikipedia, the free encyclopedia |
 | | The relation of the number of combinations to the coefficients occurring in the binomial expansion was noted by Pingala in the 3rd century BC in a musical composition. |
 | | Al-Farisi saw the relation between polygonal numbers and the binomial coefficients and he presented arguments, using an early form of mathematical induction, which showed a relation between triangular numbers, the sums of triangular numbers, the sums of the sums of triangular number, etc., and the combinations of n objects taken k at a time. |
 | | For example, let f(n) be the number of distinct subsets of the integers in the interval [1,n] that do not contain two consecutive integers; e.g., with n = 4, we have the sets {}, {1}, {2}, {3}, {4}, {1,3}, {1,4}, {2,4}, so f(4) = 8. |
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