Generalized say |S| = 91 so Counting Rules |lineups of 5 students| - PowerPoint PPT Presentation

Mathematics for Computer Science Generalized Product Rule MIT 6.042J/18.062J # lineups of 5 students in class let S::= students Generalized say |S| = 91 so Counting Rules |lineups of 5 students| = 91 5 ? NO! lineups have no repeats: |seqs in S 5 with no repeats| ? Albert R Meyer, April 19, 2013 Albert R Meyer, April 19, 2013 genprod.1 genprod.2 Generalized Product Rule Generalized Product Rule |seqs in S 5 with no repeats| Q a set of length-k sequences 91 choices for 1 st student, if n 1 possible 1 st elements, 90 choices for 2nd student, n 2 possible 2 nd elements 89 choices for 3rd student, (for each first entry), n 3 possible 3 rd elements 88 choices for 4th student, (for each 1 st & 2 nd entry,…) 87 choices for 5th student 91! then, |Q| = n � 1 n ��� n k = 91 � 90 � 89 � 88 � 87 = 2 86! Albert R Meyer, April 19, 2013 Albert R Meyer, April 19, 2013 genprod.3 genprod.4 1

Division Rule Division Rule if function from A to B #6.042 students = is k-to-1, then #6.042 students' fingers |A| = k|B| 10 (generalized Bijection Rule) Albert R Meyer, April 19, 2013 Albert R Meyer, April 19, 2013 genprod.5 genprod.6 Counting Subsets Counting Subsets How many size 4 subsets of {1,2,…,13}? a 1 a 3 a 2 a 4 a 5 …a 12 a 13 also maps Let A::= permutations of {1,2,…,13} to {a 1 , a 2 , a 3 , a 4 } B::= size 4 subsets so does a 1 a 3 a 2 a 4 a 13 … a 12 a 5 map a 1 a 2 a 3 a 4 a 5 …a 12 a 13 ∈ A 4! perms 9! perms to {a 1 , a 2 , a 3 , a 4 } ∈ B all map to same set 4! � 9!-to-1 Albert R Meyer, April 19, 2013 Albert R Meyer, April 19, 2013 genprod.7 genprod.8 2

Counting Subsets Counting Subsets # m element subsets 13! = |A| = (4! � 9!)|B| of an n element set is so # of size 4 subsets is   n n!  13  13! ::= m!(n - m)!   =   :: m   4!9! 4 �   n choose m Albert R Meyer, April 19, 2013 Albert R Meyer, April 19, 2013 genprod.9 genprod.10 3

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