Compute the number of derangements in [math]S_4[/math], by explicitely listing them, and then comment on the estimate of

that you obtain in this way.

Show that the probability for a length [math]1[/math] needle to intersect, when thrown, a [math]1[/math]-spaced grid is [math]2/\pi[/math], and then comment on the estimate on

that you obtain in this way.

Interpret the abstract PLT formula established above, namely

as a Poisson Limit Theorem, with full probabilistic details.

Find some formulae for the Bell numbers [math]B_k[/math], or rather for their generating series, or suitable transforms of that series, and the more the better.

Show that the truncated characters of [math]S_N[/math], suitably moved over the diagonal, as to not overlap, become independent with [math]N\to\infty[/math].

Find some alternative proofs for the fact, that we already know, that the truncated charcters for [math]A_N\subset O_N[/math] become Poisson, with [math]N\to\infty[/math].