Updated for extra context and clarity
I have this question that I can’t figure out and would appreciate some help. The first part of the exercise what about distinguishable particles and the co vergence to Poisson(r) was clear. However for this second part I am getting stuck even when using the Stirling approximation.
Given that $\ m = \lfloor nr \rfloor $ indistinguishable particles are distributed randomly into $\ n $ urns, and $\ X_n $ is the number of particles in urn number 1, we want to calculate the probability $\ P(X_n = k) $. The probability is given by the ratio of two combinatorial expressions:
$$ P(X_n = k) = \frac{\binom{m - k + n - 2}{m - k}}{\binom{m + n - 1}{m}} $$
The goal is to analyze this probability and show how it behaves as $\ n \to \infty $, particularly whether it converges in distribution to a geometric distribution in the limit: $\ X_n\to Geo_0(1/(1+r))$
