Stitz-Zeager_College_Algebra_e-book

# Reduced fraction multiples of with a denominator of 6

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Unformatted text preview: o decide which three friends have to stay home.) The reader is encouraged to verify this by actually taking the time to list all of the possibilities. We now state anf prove a theorem which is crucial to the proof of the Binomial Theorem. Theorem 9.3. For natural numbers n and j with n ≥ j , n n + j−1 j = n+1 j The proof of Theorem 9.3 is purely computational and uses the deﬁnition of binomial coeﬃcients, the recursive property of factorials and common denominators. 3 For reference, 50! 50! 40! 50! 40!10! = 30414093201713378043612608166064768844377641568960512000000000000, = 37276043023296000, = 10272278170 and 584 Sequences and the Binomial Theorem n n + j−1 j = n! n! + (j − 1)!(n − (j − 1))! j !(n − j )! = n! n! + (j − 1)!(n − j + 1)! j !(n − j )! = n! n! + (j − 1)!(n − j + 1)(n − j )! j (j − 1)!(n − j )! = n!(n − j + 1) n! j + j (j − 1)!(n − j + 1)(n − j )! j (j − 1)!(n − j + 1)(n − j )! = n! j n!(n − j + 1) + j !(n − j + 1)! j !(n − j + 1)! = n! j + n!(n − j + 1)...
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## This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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