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| + | ==Permutations== | ||
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| + | ==Combinations== | ||
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| + | ==Binomial Coefficients== | ||
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| + | ==Pascal's Triangle== | ||
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==Binomial Theorem== | ==Binomial Theorem== | ||
===Definition=== | ===Definition=== | ||
Revision as of 04:12, 5 September 2008
Contents
Permutations
Combinations
Binomial Coefficients
Pascal's Triangle
Binomial Theorem
Definition
$ (x+y)^n = \sum_{i=0}^n {n \choose k}x^i y^{n-i}, $
$ \text{where } {n \choose k} = \frac{n!}{n!(n-r)!}. $
Example
- What is $ \sum_{i=0}^n {n \choose k} = {n \choose 0} + {n \choose 1} + .. + {n \choose n} ? $
Solution: Using the Binomial Theorem, let x = y = 1. Then, $ \sum_{i=0}^n {n \choose k} (1)^i (1)^{n-i} = \underline{\sum_{i=0}^n {n \choose k}} = (1+1)^n = \underline{2^n}. $
