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Math119PascalMagic

# Math119PascalMagic - THE MAGIC OF PASCAL'S TRIANGLE...

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THE MAGIC OF PASCAL'S TRIANGLE PASCAL'S TRIANGLE This represents a way to write down the "early" binomial coefficients n r Ê Ë Á ˆ ¯ ˜ easily. • Each row begins and ends with "1". (We have a "tent" of "1"s.) • Every other entry equals the sum of the two entries immediately above it. Here it is: 1 Row 0: Contains 0 0 Ê Ë Á ˆ ¯ ˜ 1 1 Row 1: Contains 1 0 1 1 Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ , 1 2 1 Row 2: Contains 2 0 2 1 2 2 Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ , , 1 3 3 1 Row 3: Contains 3 0 3 1 3 2 3 3 Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ , , , 1 4 6 4 1 Row 4: Contains 4 0 4 1 4 2 4 3 4 4 Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ , , , , etc. - The "histograms" of the rows approach a bell-shaped "normal" distribution! We can observe some basic properties of binomial coefficients: Symmetry about the center: n r n n r Ê Ë Á ˆ ¯ ˜ = - Ê Ë Á ˆ ¯ ˜ (The process of choosing r winners is equivalent to the process of choosing n-r losers.) The "tent" of "1"s: n n n 0 1 Ê Ë Á ˆ ¯ ˜ = Ê Ë Á ˆ ¯ ˜ =

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