The probability of a successful allocation of ball groups by boxes

(Lobachevskii Journal of Mathematics, Volume XXV)

Let p=p_Nn be the probability of a successful allocation of n groups of distinguishable balls in N boxes in equiprobable scheme for group allocation of balls with the following assumption: each group contains m balls and each box contains not more than q balls from a same group. If q=1, then we easily calculate p and observe that p→ e^{-α₀(m-1)/2} as n, N→∞ such that α = n/N → α₀<∞. In the case 2 ≤ q we also find an explicit formula for the probability and prove that p→ 1 as n, N→∞ such that α=n/N ≤ α'<∞.