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lec-16 - CS70 Satish Rao Lecture 16 1 Review 2 Stars/Bars 3...

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CS70: Satish Rao: Lecture 16. 1. Review. 2. Stars/Bars. 3. Balls in Bins. 4. Combinatorial Proofs. 5. Inclusion/Exclusion
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First rule: n 1 × n 2 ···× n 3 . Product Rule.
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First rule: n 1 × n 2 ···× n 3 . Product Rule. k Samples with replacement from n items: n k .
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First rule: n 1 × n 2 ···× n 3 . Product Rule. k Samples with replacement from n items: n k . Sample without replacement: n ! ( n - k )!
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First rule: n 1 × n 2 ···× n 3 . Product Rule. k Samples with replacement from n items: n k . Sample without replacement: n ! ( n - k )! Second rule: when order doesn’t matter divide..when possible.
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First rule: n 1 × n 2 ···× n 3 . Product Rule. k Samples with replacement from n items: n k . Sample without replacement: n ! ( n - k )! Second rule: when order doesn’t matter divide..when possible. Sample without replacement and order doesn’t matter: ( n k ) = n ! ( n - k )! k ! . n choose k
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Stars and bars .... How many ways can Bob and Alice split 5 dollars?
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Stars and bars .... How many ways can Bob and Alice split 5 dollars? For each of 5 dollars pick Bob or Alice(2 5 ), divide out order
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Stars and bars .... How many ways can Bob and Alice split 5 dollars? For each of 5 dollars pick Bob or Alice(2 5 ), divide out order ???
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Stars and bars .... How many ways can Bob and Alice split 5 dollars? For each of 5 dollars pick Bob or Alice(2 5 ), divide out order ??? 5 dollars for Bob and 0 for Alice:
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Stars and bars .... How many ways can Bob and Alice split 5 dollars? For each of 5 dollars pick Bob or Alice(2 5 ), divide out order ??? 5 dollars for Bob and 0 for Alice: one unordered set: ( B , B , B , B , B ).
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Stars and bars .... How many ways can Bob and Alice split 5 dollars? For each of 5 dollars pick Bob or Alice(2 5 ), divide out order ??? 5 dollars for Bob and 0 for Alice: one unordered set: ( B , B , B , B , B ). 4 for Bob and 1 for Alice:
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Stars and bars .... How many ways can Bob and Alice split 5 dollars? For each of 5 dollars pick Bob or Alice(2 5 ), divide out order ??? 5 dollars for Bob and 0 for Alice: one unordered set: ( B , B , B , B , B ). 4 for Bob and 1 for Alice: 5 unordered sets: ( A , B , B , B , B ) ; ( B , A , B , B , B ); ...
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Stars and bars .... How many ways can Bob and Alice split 5 dollars? For each of 5 dollars pick Bob or Alice(2 5 ), divide out order ??? 5 dollars for Bob and 0 for Alice: one unordered set: ( B , B , B , B , B ). 4 for Bob and 1 for Alice: 5 unordered sets: ( A , B , B , B , B ) ; ( B , A , B , B , B ); ... Single way to specify, first Alice’s dollars, then Bob’s. ( B , B , B , B , B ) ( A , B , B , B , B ) ( A , A , B , B , B ) and so on.
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Stars and bars .... How many ways can Bob and Alice split 5 dollars? For each of 5 dollars pick Bob or Alice(2 5 ), divide out order ??? 5 dollars for Bob and 0 for Alice: one unordered set: ( B , B , B , B , B ). 4 for Bob and 1 for Alice: 5 unordered sets: ( A , B , B , B , B ) ; ( B , A , B , B , B ); ... Single way to specify, first Alice’s dollars, then Bob’s. ( B , B , B , B , B ) ( A , B , B , B , B ) ( A , A , B , B , B ) and so on. Counting these.. 0 + 5, 1 + 4,2 + 3, 3 + 2, 4 + 1, 5 + 0.
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Stars and bars .... How many ways can Bob and Alice split 5 dollars?
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