Combinatorics How to Count Lecture 6/3
The die is cast. Methods of Counting Combinatorics (coined circa 1951) refers to the methods used to count things. Why combinatorics? If a sample spaces contains a finite set of outcomes, determining the probability of an event often is a counting problem. But often the numbers are just too large to count in the 1, 2, 3, 4 ordinary way. For example, if you put a grain of rice on the first square of a chessboard, then two grains on the second square, four on the third square, and continue doubling until all 64 squares are filled, how many grains of rice would you have in all? This number although rather small, represents more rice than has ever been harvested in the history of mankind. If you place the grains end to end they would line up to be 7.8 light years! Yet we are expected to handle these kinds of numbers. Since counting methods are needed in the sequel, we begin with a review of counting methods. Many of these theorems can be illustrated using the Combinatorial Object Server. As an application of the counting theorems, let's establish some improtant and useful connections between the number of combinations of n distinct objects taken r at a time, binomial coefficients and Pascal's Triangle.
Stirling's Formula
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