Honors Discrete Mathematics
Honors Discrete Mathematics MAD 2104
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This 4 page Class Notes was uploaded by Dino Corwin on Monday October 12, 2015. The Class Notes belongs to MAD 2104 at Florida Atlantic University taught by Jorge Viola-Prioli in Fall. Since its upload, it has received 17 views. For similar materials see /class/221637/mad-2104-florida-atlantic-university in Mathematics Discrete at Florida Atlantic University.
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Date Created: 10/12/15
16BINOMIAL COEFFICIENTS In section 9 we showed that there are 2n subsets of any set with n elements As a consequence we computed the number of subsets with at most 18 elements of a set with 20 elements You are invited to review the solution we presented then before proceeding There was a reason not to try to solve the similar problem with at most 17 elements we need a formula to calculate how many sets with exactly 17 elements are there in a 20element set 50 we address now the general problem if A has n elements how many subsets ofA with exactly k elements are there Of course here k s n n The symbol k will denote the number of kelement subsets ofan nelement set for O s k s n J Viola Prioli n is called a binomial coefficient for reasons to be seen later and is also called quotn choose kquot for obvious reasons Calculators and some textbooks n have an alternative symbol for J that IS nCk Please be adVIsed that we k will use them indistinctly throughout Before proceeding let us underscore the difference between lists of length two with or without repetitions and 2element sets out of a b c There are 23 subsets 8 subsets There are 9 lists of length 2 There are 6 lists of length 2 if repetitions are not allowed There are 3 2element sets a b a c and b c J Viola Prioli Let us return to the de nition ofthe combinatorials It follows from it that n a each k isa natural number A formula to compute is next n n THEOREM k k n k Proof t suf ces to prove that x k x n k n J Viola Prioli