CHAPTER 11 FurtherTopics in Algebra Evaluate 5C3. 5C3 = 5! 3!15 - 32! = 5! 3! 2! = 5 # 4 # 3 # 2 # 1 3 # 2 # 1 # 2 # 1 = 10 Expand 12m+ 324. 12m+ 324 = 12m24 + 4! 1! 3! 12m23132 + 4! 2! 2! 12m221322 + 4! 3! 1! 12m21323 + 34 = 24m4 + 41223m3132 + 61222m2192 + 412m21272 + 81 = 16m4 + 12182m3 + 54142m2 + 216m+ 81 = 16m4 + 96m3 + 216m2 + 216m+ 81 Find the eighth term of the binomial expansion of 1a - 2b210. a10 7 ba31-2b27 = 120a31-128b72 = -15,360a3b7 Binomial Coefficient For nonnegative integers n and r, with r … n, nCr = a n r b = n! r!1n −r2! . Binomial Theorem For any positive integer n and any complex numbers x and y, 1x + y2n is expanded as follows. 1x +y2n =xn + a n 1b xn−1y +a n 2b xn−2y2 +a n 3b xn−3y3 + P + a n r b xn−ryr +P+ a n n −1b xyn−1 +yn k th Term of the Binomial Expansion of 1x +y2n a n k −1b xn−1k−12yk−1 1where n #k −12 See Examples 1 and 2 in this section. Example 3 in this section illustrates the generalized principle of mathematical induction. If there are 2 ways to choose a pair of socks and 5 ways to choose a pair of shoes, then how many ways are there to choose socks and shoes? 2 # 5 = 10 ways How many ways are there to arrange the letters of the word TRIANGLE using 5 letters at a time? This is an arrangement. Use permutations. P18, 52 = 8! 18 - 52! = 8! 3! = 6720 Let n = 8 and r = 5. 1114 11.5 Mathematical Induction Principle of Mathematical Induction Let Sn be a statement concerning the positive integer n. Suppose that both of the following are satisfied. 1. S1 is true. 2. For any positive integer k, k … n, if Sk is true, then Sk+1 is also true. Then Sn is true for every positive integer value of n. 11.6 Basics of Counting Theory Fundamental Principle of Counting If n independent events occur, with m1 ways for event 1 to occur, m2 ways for event 2 to occur, c, and mn ways for event n to occur, then there are m1 # m2 # P# mn different ways for all n events to occur. Permutations Formula If P1n, r2 denotes the number of permutations of n elements taken r at a time, with r … n, then the following holds true. P1n, r2 = n! 1 n −r2! Concepts Examples
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