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College Algebra Exam Review 50

# College Algebra Exam Review 50 - 60 1 ALGEBRAIC THEMES...

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Unformatted text preview: 60 1. ALGEBRAIC THEMES Corollary 1.9.6. n X ÂnÃ n (a) 2 D . k k D0 ÂÃ n X kn (b) 0 D . 1/ . k k D0 n n X ÂnÃ X ÂnÃ n1 (c) 2 D D . k k kD0 kD0 k odd k even Proof. Part (a) follows from the combinatorial interpretation of the two sides: The total number of subsets of an n element set is the sum over k of the number of subsets with k elements. Part (a) also follows from the binomial theorem by putting x D y D 1. Part (b) follows from the binomial theorem by putting x D 1; y D 1. The two sums in part (c) are equal by part (b), and they add up to 2n by part (a); hence each is equal to 2n 1 . I Example 1.9.7. We can obtain many identities for the binomial coefﬁcients by starting with the special case of the binomial theorem: n X ÂnÃ n .1 C x/ D xk ; k k D0 regarding both sides as functions in a real variable x , manipulating the functions (for example, by differentiating, integrating, multiplying by x , etc.), and ﬁnally evaluating for a speciﬁc value of x . For example, differentiating the basic formula gives n X ÂnÃ n.1 C x/n 1 D k xk 1: k k D1 Evaluating at x D 1 gives n X ÂnÃ D k ; k n1 n2 k D1 while evaluating at x D 1 gives 0D n X . 1/ k D1 k1 ÂÃ n k : k ...
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