Probability MT22, Misc
Flashcards
Counting and binomial identities
@State Stirling’s formula for an approximation of $n!$.
\[n! \sim \sqrt{2\pi} n^{n+1/2} e^{-n}\]
@State Vandermonde’s identity for
\[\binom{m+n}{k}\]
\[\sum^k _ {j=0} \binom{m}{j} \binom{n}{k-j}\]
@Prove Vandermonde’s identity (∆vandermondes-identity), i.e.
\[\binom{m+n}{k} = \sum _ {j=0}^k \binom{m}{j} \binom{n}{k-j}\]
@todo (probability, page 6).
What’s the intuitive reason ${}^n C _ k = {}^n C _ {n-k}$?
Because choosing $k$ objects is the same as not choosing the other $n - k$ objects.
Countability
@Define what it means for a set $S$ to be countable.
Either there exists a bijection $\mathbb{N} \to S$ or $S$ is finite.
Methods
@Describe the stars and bars method.
Solving combinatorial problems involving bins by considering the permutations of stars and bars.