1074 CHAPTER 11 Further Topics in Algebra For example, in row four of the triangle, 1 is the sum of 1 (the only number above it), 4 is the sum of 1 and 3, 6 is the sum of 3 and 3, and so on. This triangular array of numbers is called Pascal’s triangle, in honor of the 17th-century mathematician Blaise Pascal. It was, however, known long before his time. To find the coefficients for 1x + y26, we need to include row six in Pascal’s triangle. Adding adjacent numbers in row five, we find that row six is 1 6 15 20 15 6 1. Using these coefficients, we obtain the expansion of 1x + y26. 1x + y26 = x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6 n-Factorial For any positive integer n, n! =n1n −12 1n −22 P132 122 112. By definition, 0! =1. For example, 5! = 5 # 4 # 3 # 2 # 1 = 120, 7! = 7 # 6 # 5 # 4 # 3 # 2 # 1 = 5040, and 2! = 2 # 1 = 2. Blaise Pascal (1623–1662) Pascal, a French mathematician, made mathematical contributions in the areas of calculus, geometry, and probability theory. At age 19, he invented the first adding machine, a precursor to our modern-day calculator. n-Factorial Although it is possible to use Pascal’s triangle to find the coefficients of 1x + y2n for any positive integer n, this calculation becomes impractical for large values of n because of the need to write all the preceding rows. A more efficient way of finding these coefficients uses factorial notation. The number n! (read “n-factorial ”) is defined as follows. Binomial Coefficients Now look at the coefficients of the expansion 1x + y25 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5. The coefficient of the second term, 5x4y, is 5, and the exponents on the variables are 4 and 1. Note that 5 = 5! 4! 1! . The coefficient of the third term, 10x3y2, is 10, with exponents of 3 and 2 on the variables, and 10 = 5! 3! 2! .
RkJQdWJsaXNoZXIy NjM5ODQ=