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Mathematics for Computer Science MIT 6.042J/18.062J Polynomials Express Choices & Outcomes ( ) ( ) Binomial = + + + Theorem + + + + + Image by MIT OpenCourseWare. Products of Sums = Sums of Products Albert R Meyer, April 18, 2012 Albert R Meyer, April 18, 2012 lec 10W.1 lec 10W.2 expression for c k ? binomial expressions (1+X) 0 = 1 (1+X) n = (1+X) 1 = 1 + 1X 1 + 2X + 1X 2 (1+X) 2 = 2 n c 0 +c 1 X+c 2 X +...+c n X (1+X) 3 = 1 + 3X + 3X 2 + 1X 3 (1+X) 4 = 1 + 4X + 6X 2 + 4X 3 + 1X 4 Albert R Meyer, April 18, 2012 Albert R Meyer, April 18, 2012 lec 10W.3 lec 10W.4 1

expression for c k ? expression for c k ? (1 + X) n (1 + X) n n times n times = (1 + X)(1 + X)(1 + X)(1 + X)...(1 + X) = (1 + X)(1 + X)(1 + X)(1 + X)...(1 + X) the X k coeff, c k , is # terms multiplying gives 2 n product terms: with exactly k X’s selected 11 ⋯ 1 + X11X ⋯ X1 + 1XX ⋯ 1X1 + ⋯ + XX ⋯ X ⎛ ⎞ n a term corresponds to selecting 1 or X c = ⎜ ⎟ from each of the n factors k k ⎝ ⎠ Albert R Meyer, April 18, 2012 Albert R Meyer, April 18, 2012 lec 10W.5 lec 10W.6 The Binomial Formula The Binomial Formula binomial (X + Y) = n (1 + X) n = expression ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ n n n ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ n n n n n n n-1 2 n-2 Y + XY + ⎟ X Y + ⎜ ⎟ ⎜ ⎟ ⎜ ... + ⎜ ⎟ X 2 X k n + X + X + ... + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 1 2 0 1 2 k n ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ n n k n-k + ... + ⎜ ⎟ X n ... + ⎜ ⎟ X Y k n ⎝ ⎠ ⎝ ⎠ binomial coefficients Albert R Meyer, April 18, 2012 Albert R Meyer, April 18, 2012 lec 10W.7 lec 10W.8 2

The Binomial Formula ⎛ n ⎞ n = ∑ ⎜ (X + Y) n ⎟ X k Y n-k k=0 ⎝ k ⎠ Albert R Meyer, April 18, 2012 lec 10W.9 3

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