Assume you have a $N$ balls. You give each ball $T$ different labels randomly from $\{0,\dots, N-1\}$. So hamming weight of each of labelling varies from $0$ to $\lceil\log_2 N\rceil$. What is probability $P(h)$ that a given ball has label with Hamming weight greater than $h$ in each of $T$ labels? I am looking for asymptotics.
Also at what minimum weight $w_p$ does probability $$\sum_{h=0}^{w_p}P(h)>p\in(0,1)$$ hold?