Note: You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes from 2003 to 2012. Many mathematical formulas are broken, and there are likely to be other bugs as well. These will most likely not be fixed. You may be able to find more up-to-date versions of some of these notes at http://www.cs.yale.edu/homes/aspnes/#classes.

• More examples of product rule
• Permutations: number of bijections from [n] to [n] is n! = n(n-1)(n-2)...1.
• k-permutations: number of injections from [k] to [n] is n(k) = n(n-1)...(n-k+1)

• E.g. counting the number of sequences of 5 cards we can deal from a 52-card deck
• But what if we don't care about order?
• Counting two ways
• C(n,k) [n choose k] = |{A⊆S: |A| = k}|
• We'll count these by connecting them to P(n,k)
• Method 1: P(n,k) = n(k) (already done)

• Method 2: P(n,k) = C(n,k)⋅k! (choose the subset then order it)
• So P(n,k) = C(n,k)⋅k! ⇒ C(n,k) = n(k)/k! = n!/(k!(n-k)!)

• Reducing to previous case
• n identical balls in k bins
• No restrictions: place k-1 dividers among n+k-1 spaces to get (n+k-1 choose k-1)
• All bins nonempty: remove k balls and reduce to previous case
• Binomial coefficients
• Definition in terms of factorials: already given
• Comes from binomial theorem
• Recursive definition using Pascal's identity: (n choose k) = (n-1 choose k) + (n-1 choose k-1)
• Easy combinatorial proof
• Painful algebraic proof
• Algebraic identity proofs are an example of GeneratingFunctions

2014-06-17 11:57