Suppose there are $N$ people in a party. Each of them brings $k$ gifts. When the party is over, each of them takes $k$ gift randomly. Denote $T$ is the number of gifts return to its original giver. Please find the limiting distribution with $N\to +\infty$ and $k$ fixed.
From this link Generalized Derangement Problem with $k$ unchanged elements , I find the generating function $$f(u)=\lim_{N\to \infty}\int_{0}^{+\infty}\frac{(k!)^N}{(Nk)!}(u-1)^{kN}L_k^N(\frac{x}{1-u})e^{-x}dx$$
However, I have trouble evaluate the coefficients for general $k$. I don't know much about the Laguerre polynomials.
Thanks in advance.
