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# lec06 - AMS 301 Lecture 6 AMS Ning SUN Aug 4 2009...

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AMS 301 Lecture 6 Ning SUN Aug 4 2009

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Arrangements and Selections with Repetitions
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? AMS301, Summer 2009, Ning SUN

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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? AMS301, Summer 2009, Ning SUN
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? AMS301, Summer 2009, Ning SUN

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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 ! AMS301, Summer 2009, Ning SUN
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 ! AMS301, Summer 2009, Ning SUN

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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
Brain Teaser (cont’d) 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 AMS301, Summer 2009, Ning SUN

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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 AMS301, Summer 2009, Ning SUN
Theorem2 The number of selections with repetition of r objects chosen from n types of objects is: C ( r + n - 1, r ) AMS301, Summer 2009, Ning SUN

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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 . How many different ways to select 4 cup of ice cream, 1 cup for each of them? Repetition allowed Order matters 3 3 3 3 = 81 AMS301, Summer 2009, Ning SUN
If the question is changed again… Q5: You want to buy ice cream from the store, and there are three flavors: vanilla , chocolate and strawberry . How many different ways to select 6 cups of ice cream with the requirement that at least one cup of each flavor is picked ?

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lec06 - AMS 301 Lecture 6 AMS Ning SUN Aug 4 2009...

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