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Albert R Meyer, April 18, 2012

Mathematics for Computer Science

MIT 6.042J/18.062J

Binomial Theorem

lec 10W.1 Albert R Meyer, April 18, 2012

Polynomials Express Choices & Outcomes

Products of Sums = Sums of Products

lec 10W.2 Albert R Meyer, April 18, 2012

expression for ck?

(1+X)n = c

2 n 0 +c 1X+c2X +...+cnX

lec 10W.3 Albert R Meyer, April 18, 2012

binomial expressions (1+X)0 = 1 (1+X)1 = 1 + 1X (1+X)2 = 1 + 2X + 1X2 (1+X)3 = 1 + 3X + 3X2 + 1X3 (1+X)4 = 1 + 4X + 6X2 + 4X3 + 1X4

lec 10W.4

( ( ) )

+

=

+ + + + + + +

Image by MIT OpenCourseWare.

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Albert R Meyer, April 18, 2012

(1 + X)n

n times

= (1 + X)(1 + X)(1 + X)(1 + X)...(1 + X) multiplying gives 2n product terms:

11⋯1 + X11X⋯X1 + 1XX⋯1X1 +⋯+ XX⋯ X

a term corresponds to selecting 1 or X from each of the n factors

expression for ck?

lec 10W.5 Albert R Meyer, April 18, 2012

the Xk coeff, ck, is # terms with exactly k X’s selected

n ⎛ ⎞

k

c =⎜ ⎟ k ⎝ ⎠

(1 + X)n

n times

= (1 + X)(1 + X)(1 + X)(1 + X)...(1 + X) expression for ck?

lec 10W.6 Albert R Meyer, April 18, 2012

The Binomial Formula

⎛ ⎞ ⎛ ⎞ n n ⎛ ⎞

2

n ⎛ ⎞ n + X + X + ... + Xk ⎛ ⎞ +

n

n ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ... + ⎜ ⎟X 1 2 k n ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

binomial coefficients

binomial

(1 + X)n =

expression

lec 10W.7 Albert R Meyer, April 18, 2012

The Binomial Formula

⎛ ⎞ ⎛ ⎞ n n ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜

n n-1

n Y + XY + ⎟X Y +

2 n-2

⎝ ⎠ ⎝ ⎠ 1 ⎝ ⎠ 2 ⎛ ⎞ n n ... + X Y

k n-k

⎛ ⎞ ⎜ ⎟ + ... + ⎜ ⎟Xn k n ⎝ ⎠ ⎝ ⎠

n

(X + Y) =

lec 10W.8

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Albert R Meyer, April 18, 2012

The Binomial Formula

n

(X + Y)n ⎛ n ⎞ = ∑⎜ ⎟ XkYn-k

k=0⎝ k ⎠

lec 10W.9

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