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ics141-lecture24-Combinatorics2

# ics141-lecture24-Combinatorics2 - University of Hawaii...

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24-1 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii ICS141: Discrete Mathematics for Computer Science I Department of Information and Computer Sciences University of Hawaii Stephen Y. Itoga

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24-2 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Lecture 24 Chapter 5. Counting 5.4 Binomial Coefficients 5.5 Generalized Permutations and Combinations
24-3 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Some material in these slides were taken/adapted from the slides made by Dr. Pomplun Some slides were done by Prof. Baek

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24-4 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Binomial Coefficients Expressions of the form C ( n , r ) are also called binomial coefficients Coefficients of the expansion of powers of binomial expressions Binomial expression is a simply the sum of two terms such as x+y Example: 3 2 2 3 3 2 2 3 3 3 3 ) 3 , 3 ( ) 2 , 3 ( ) 1 , 3 ( ) 0 , 3 ( ) )( ( ) )( )( ( ) ( y xy y x x y C xy C y x C x C yyy yyx yxy yxx xyy xyx xxy xxx y x yy yx xy xx y x y x y x y x + + + = + + + = + + + + + + + = + + + + = + + + = +
24-5 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii The Binomial Theorem Let x and y be variables, and let n be a nonnegative integer. Then To obtain a term of the form x n-j y j , it is necessary to choose n-j x’ s from the n sums, so that the other j terms in the product are y’ s. Therefore, the coefficient of x n-j y j is C ( n , n-j ) = C ( n , j ). The binomial theorem gives the coefficients of the expansion of powers of binomial expressions. ( 29 n n n n n j j n n j n y n n xy n n y x n y x n x n y x j n y x + - + + + + = = + - - - - = 1 2 2 1 0 1 2 1 0

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24-6 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Examples ( a+b ) 9 the coefficient of a 5 b 4 = C (9,4) The coefficient of x 12 y 13 in the expansion of (2 x– 3 y ) 25 By binomial theorem The coefficient of x 12 y 13 is obtained when j = 13 ( x+y+z ) 9 the coefficient of x 2 y 3 z 4 = C (9,2)· C (7,3) ( 29 j j j y x j y x ) 3 ( ) 2 ( 25 ) 3 ( 2 25 25 0 25 - = - + - = 13 12 13 12 3 2 ! 12 ! 13 ! 25 ) 3 ( 2 ) 13 , 25 ( - = - C
24-7 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii T Proof: T Proof: It implies that: integer. positive a is where , n k n n k k 0 ) 1 ( 0 = - = Corollaries integer. e nonnegativ a is where n k n n n k , 2 0 = = - = - = - + = = = = - k n k n n k k n k k k n n n 0 0 ) 1 ( ) 1 ( 1 )) 1 ( 1 ( 0 0 = - = = = + = n k k k n n k n n k n k n 0 0 1 1 ) 1 1 ( 2 + + + = + + + 5 3 1 4 2 0 n n n n n n

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24-8 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii T Proof: Corollaries (cont.) integer. e nonnegativ a is where , n k n n n k k 3 2 0 = = k n k k k n n k n n k n k n 2 2 1 ) 2 1 ( 3 0 0 = - = = = + =
24-9

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ics141-lecture24-Combinatorics2 - University of Hawaii...

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