{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–17. Sign up to view the full content.

CS70: Satish Rao: Lecture 16. 1. Review. 2. Stars/Bars. 3. Balls in Bins. 4. Combinatorial Proofs. 5. Inclusion/Exclusion

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
First rule: n 1 × n 2 ···× n 3 . Product Rule.
First rule: n 1 × n 2 ···× n 3 . Product Rule. k Samples with replacement from n items: n k .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 )!
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Stars and bars .... How many ways can Bob and Alice split 5 dollars?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 ???

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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:
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 ).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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:
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 ); ...

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Stars and bars .... How many ways can Bob and Alice split 5 dollars?
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}