lec06 - AMS 301 Lecture 6 AMS 301 Lecture 6 Ning SUN Aug 4...

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Unformatted text preview: AMS 301 Lecture 6 AMS 301 Lecture 6 Ning SUN Aug 4 2009 Arrangements and Arrangements and Selections with Repetitions Selections with Repetitions Warmup Warmup Q0: In your fridge, there are four cups of ice cream of different flavors: vanilla , chocolate, peppermint and strawberry . You want to pick up 2 cups. How many different ways can you make it? 3 AMS301, Summer 2009, Ning SUN Warmup Warmup Q1: In your fridge, there are four cups of ice cream of different flavors: vanilla , chocolate, peppermint and strawberry . You want four kids: John, Mary, Rose and Steve, to get one cup per person. How many different arrangements? 4 AMS301, Summer 2009, Ning SUN Warmup Warmup Q2: Today you only have two flavors: two cups of vanilla and two cups of strawberry . You want four kids: John, Mary, Rose and Steve, to get one cup per person. How many different arrangements? 5 AMS301, Summer 2009, Ning SUN Theorem1 Theorem1 If there are n objects, with r 1 of type 1, r 2 of type 2, , and r m of type m , where r 1 + r 2 + + r m = n , then the number of arrangements of these n objects, denoted P ( n ; r 1 , r 2 ,, r m ), is P ( n ; r 1 , r 2 ,, r m )= C ( n , r 1 ) C ( n-r 1 , r 2 ) C ( n-r 1-r 2 , r 3 ) C ( n-r 1--r m-1 , r m ) = n ! r 1 ! r 2 !... r m ! 6 AMS301, Summer 2009, Ning SUN Look at the formula Look at the formula C ( n , r 1 ) C ( n-r 1 , r 2 ) C ( n-r 1-r 2 , r 3 ) C ( n-r 1--r m-1 , r m ) = = n ! r 1 !( n-r 1 )! ( n-r 1 )! r 2 !( n-r 1-r 2 )! ( n-r 1-r 2 )! r 3 !( n-r 1-r 2-r 3 )! ( n-r 1--r m-1 )! r m !( n-r 1--r m )! r 1 + r 2 + + r m = n ( n-r 1--r m )! =1 n ! r 1 ! r 2 !... r m ! 7 AMS301, Summer 2009, Ning SUN Brain Teaser Brain Teaser Q3: You have to buy some ice cream from the store, and there are three flavors: vanilla , chocolate and strawberry . How many different ways to select 4 cups of ice cream? xx s x s x xx s xx s xxx ss x Repetition allowed Order does NOT matter 8 Brain Teaser Brain Teaser (contd) (contd) There is a 1-to-1 correspondence between selections and such sequences How many different ways to arrange 4 x s and 2 s s xx s x s x 9 AMS301, Summer 2009, Ning SUN The question is equivalent to The question is equivalent to There are 6 objects, with 4 of type 1 and 2 of type 2. How many different arrangements can you make? By theorem 1: C(6,4) C(2,2) = 6!/(4!2!) = 15 xxxx ss xxx s x s xxx ss x xx s xx s xx ss xx x s xxx s x ss xxx x s xx s x x s x s xx xx s x s x s xxx s x s xx s xx s x s xxx ss xxxx s xxxx s 1 AMS301, Summer 2009, Ning SUN Theorem2 Theorem2 The number of selections with repetition of r objects chosen from n types of objects is: C ( r + n - 1, r ) 1 1 AMS301, Summer 2009, Ning SUN If the question is changed If the question is changed Q4: You take the four kids (John, Mary, Rose and Steve) to buy ice cream from the store, and there are three flavors: vanilla , chocolate and strawberry ....
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This note was uploaded on 08/30/2010 for the course AMS 301 taught by Professor Arkin during the Spring '08 term at SUNY Stony Brook.

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lec06 - AMS 301 Lecture 6 AMS 301 Lecture 6 Ning SUN Aug 4...

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