By Gottfried Wilhelm Leibniz

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Y + z) 8? 5. cl' where a, b, c range over all non-negative integers satisfying a + b + c = 12? 6 Pascal's Triangle The binomial coefficients in the expansion of (x + y)n form an interesting pattern if listed with increasing values of n. We begin with (x + y)O = 1 to give symmetry to the table: + :)1)0 (x + :)1)1 (x + :)1)2 (x + :)1)3 1 from (x 1 1 from 1 2 1 from 1 3 3 1 1 4 6 4 from 1 1 5 10 10 5 etc. 1 1 6 15 20 15 6 1 1 7@@35 21 7 1 8 28 ~ 1 70 56 28 8 1 This array, listed here as far as n = 8, is called Pascal's triangle.

7. 2) involved consideration of a special one, T, of the n things. Consider now two special ones, say Sand T. The combinations can be divided into four classes: those that contain both Sand Tj those that contain S but not Tj those that contain T but not Sj those that contain neither S nor T. What formula results if we write C(n,r) as a sum of the numbers of members of these four classes? 2). 8. 2) holds for all pairs of integers nand 1', positive, negative or zero. What is this one exception? 4 The Binomial Expansion Any sum of two unlike symbols, such as x + y, is called a binomial.

9 in this series, p. 80.