l03_combperm

# l03_combperm - Permutations & Combinations Reading: Ross,...

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02/02/2009 CS206 - Intro. to Discrete Structures II 1 Permutations & Combinations Reading: Ross, Ch 1., Sec. 3 and 4. Rosen, Ch 5., Sec. 3 Adapted from Detlef Ronneburger Monday, February 2, 2009

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02/02/2009 CS206 - Intro. to Discrete Structures II 2 Words from letters How many 7-letter “words” (combinations of letters) can be formed from letters: {A,D,E,I,N,V,T}? ADEINVT, ADEINTV, ADEIVNT, … |{A,D,E,I,N,V,T}| = 7 ) #”words” = 7 ¢ 6 ¢ 5 ¢ 4 ¢ 3 ¢ 2 ¢ 1 = 5040 Monday, February 2, 2009
02/02/2009 CS206 - Intro. to Discrete Structures II 3 Arranging books In how many ways can one arrange five books on a shelf? Books = { b 1 , b 2 , b 3 , b 4 , b 5 } b 1 b 2 b 3 b 4 b 5 , b 1 b 2 b 3 b 5 b 4 , b 1 b 2 b 4 b 3 b 5 , … |Books| = 5 #Shelf orderings = 5 ¢ 4 ¢ 3 ¢ 2 ¢ 1 = 120 Monday, February 2, 2009

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02/02/2009 CS206 - Intro. to Discrete Structures II 4 Arranging books (cont’d) Shelf (slot) order: (S 1 ,S 2 ,S 3 ,S 4 ,S 5 ) Book set: {b 1 ,b 2 ,b 4 ,b 3 ,b 5 } Book-to-shelf-slot assignment: (S 1 =b 1 ,S 2 =b 2 ,S 3 =b 4 ,S 4 =b 3 ,S 5 =b 5 ) One can place any of the five books (e.g., b 1 ) in S 1 , then any of the remaining four (e.g., b 2 ) in S 2 , etc. Monday, February 2, 2009
02/02/2009 CS206 - Intro. to Discrete Structures II 5 Arranging books (cont’d) S 1 S 2 S 3 S 4 S 5 b 1 b 2 b 3 b 4 b 5 Books Shelf slots S 2 S 3 S 4 S 5 b 1 b 2 b 3 b 4 b 5 S 3 S 4 S 5 b 1 b 2 b 3 b 4 b 5 S 4 S 5 b 1 b 2 b 3 b 4 b 5 S 5 b 1 b 2 b 3 b 4 b 5 b 1 b 2 b 3 b 4 b 5 1 2 3 4 5 b 1 b 2 Monday, February 2, 2009

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02/02/2009 CS206 - Intro. to Discrete Structures II 6 Students entering classroom In how many ways can 30 students enter a classroom (assuming they can only enter it one at a time)? S = { s 1 , …, s 30 } set of students C = (C 1 ,…,C 30 ) order of entering classroom E.g., (C 1 =s 30 , C 2 = s 5 , …, C 30 = s 1 ) 30 ¢ 29 ¢ ¢ 2 ¢ 1 ¼ 2.652528598121910 10 32 Monday, February 2, 2009
02/02/2009 CS206 - Intro. to Discrete Structures II 7 Permutations Let there be a set of n different objects. Then, one can order (permute) the n objects in n ¢ (n-1) ¢ (n-2) ¢ ¢ 2

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## This note was uploaded on 03/24/2011 for the course CS 206 taught by Professor Fredman during the Spring '08 term at Rutgers.

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l03_combperm - Permutations & Combinations Reading: Ross,...

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