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08combination_10_v2

08combination_10_v2 - CSIS1121 Discrete Mathematics...

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1 CSIS1121 Discrete Mathematics Combinations Combinations Prof. Francis Chin, Dr SM Yiu October 7 / 8, 2010 (Chapter 5)

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2 CSIS1121 Discrete Maths Permutation (review) Permutation (review) Given a set S of n distinct objects, a permutation is an ordered arrangement of these objects . The number of permutations, P ( n , n ) = n ( n -1) … (2)(1) = n ! An r-permutation is an ordered arrangement of r objects in S . The number of r-permutations of an n -set S P ( n , r ) = n ( n -1) … ( n - r +1) = n ( n -1) … ( n - r +1) ( n-r )! / ( n-r )! = n ! / ( n-r )! To prove P ( n , r ) = P ( n , s ) P ( n - s , r - s ) by combinatorial arguments By the Product Rule, every r -permutation can be formed by an s - permutation (Task T 1 ) followed by an ( r - s )-permutation of the remaining ( n - s ) objects (Task T 2 ) Task T 1 can be done in P ( n , s ) ways; task T 2 in P ( n - s , r - s ) ways. x x x x . .. x x | x x x … x x x s elements (r-s) elements from (n-s) elements
3 CSIS1121 Discrete Maths Review - r-permutation (Rule of Sum) Review - r-permutation (Rule of Sum) (Task T 2 ) Number of r -permutations containing object w = r P ( n -1, r -1) (as w can be in r positions of any ( r -1)-permutation of the set without object w ) By Sum Rule, P ( n , r ) = P ( n -1, r ) + r P ( n -1, r -1) bcd bc w d b w cd bcd w w bcd (r-1)-permutation (r-1)-permutation r copies Without w Without w r-permutation r-permutation abcd becd bacd bced baec bdea dbec To prove P ( n , r ) = P ( n -1, r ) + r P ( n -1, r -1) by combinatorial arguments Choose an arbitrary object w in S , Divide the r -permutations into 2 sets, (Task T 1 ) Number of r -permutations not containing object w = P ( n -1, r )

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4 CSIS1121 Discrete Maths No. of Permutations w/o No. of Permutations w/o w w = = P P ( ( n n -1, -1, r r ) ) Example : S = { w, x, y, z } The 3-permutations without w (xyz), (xzy), (yxz), (zxy), (yzx), (zyx) 3 -permutations without w = 3 -permutations of { x,y,z } No. of 3 -permutations not containing object w = P(3,3) = 6 No. of r -permutations not containing object w = P ( n -1, r ) So, to verify P ( n , r ) = P ( n -1, r ) + r P ( n -1, r -1) = ( n -1)…( n - r +1)( n - r ) + r ( n -1)( n -2) …( n - r +1) = n ! / ( n - r )!
5 CSIS1121 Discrete Maths Permutations and Combinations Permutations and Combinations Let S = {A,B,C,D} 2-permutations from S = {AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC} Number of 2-permutations from S = P(4,2) = 4*3 = 12 2-combinations where the order of the elements is not important = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} Number of 2-combinations from S = C(4,2) = 6 Let S = {A,B,C,D,E} 3-combinations = {{A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E}, {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}} Number of 3-combinations = C(5,3) = 10

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6 CSIS1121 Discrete Maths r r -combination -combination An r -permutation can be obtained by the Product Rule. (Task T
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08combination_10_v2 - CSIS1121 Discrete Mathematics...

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