Probability MT22, Misc
Flashcards
How many permutations are there of $n$ distinguishable objects?
\[n!\]
@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
\[\left(\begin{matrix}m +n \\ k\end{matrix}\right)\]
\[\sum^k _ {j=0} \left(\begin{matrix}m \\ j\end{matrix}\right) \left(\begin{matrix}n \\ k -j\end{matrix}\right)\]
@Prove Vandermonde’s identity, i.e.
\[{\\,{m+n} \choose r} = \sum _ {i=0}^r {m \choose i} {n \choose r - i}\]
@todo, (probability, page 6).
What’s the intuitive reason ${}^n C _ k = {}^n C _ {n-k}$?
Because choosing $n$ objects is the same as not choosing $n - k$ objects.
What does it mean for a set $S$ to be countable?
Either there exists a bijection $\mathbb{N} \to S$ or $S$ is finite.
What’s the stars and bars method?
Solving combinatorial problems involving bins by considering the permutations of stars and bars.
What’s
\[k \left(\begin{matrix}n \\k\end{matrix}\right)\]
equivalent to?
\[n \left(\begin{matrix}n-1 \\k-1\end{matrix}\right)\]