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1
1   1
1    2    1
1    3    3    1
1   4    6    4    1
1   5    10    10    5    1
1   6    15    20    15    6    1
1   7    21    35    35    21    7    1

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You are probably familiar with Pascal's Famous Triangle as a method to generate the coefficients of the expansion of the binomial (x + y)ⁿ. But do you know why it works? Do you know that adding the numbers across a row gives a power of 2? Or that the number of times odd integers occur in any row is also a power of 2? Or that adding the numbers up a diagonal gives a Fibonacci number? We will explore these surprising patterns, and more, using counting. Counting leads to beautiful, often elementary, and always very concrete proofs. It is one of our first tools, and it is time to appreciate its full mathematical power.

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Jennifer Quinn grew up in Rhode Island. She graduated from Williams College where she studied mathematics, biology, and theater. In 1993, she received her PhD in Mathematics from the University of Wisconsin. Since then she has taught at Occidental College in Los Angeles, where she is currently Associate Professor and Chair of the Mathematics Department. In 2001, she received the distinguished teaching award from the Southern California Section of the Mathematical Association of America (MAA). Jenny is just beginning a 5-year term as co-editor of Math Horizons magazine. In addition to dozens of papers in combinatorics and graph theory, Jenny has co-authored a book, Proofs That Really Count, with acclaimed mathemagician and a past BAMA speaker Arthur Benjamin.

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