Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [T n,k ] n,k≥0 satisfying the recurrence T n,k = (b 1 k + b 2)T n−1,k−1 + [(2λb 1 + a 1)k + a 2 + λ(b 1 + b 2)]T n−1,k + λ(a 1 + λb 1)(k + 1)T n−1,k+1 , where initial conditions T n,k = 0 unless 0 ≤ k ≤ n and T 0,0 = 1. Denote by T n = n k=0 T n,k. In this paper, we show the asymptotic normality of T n,k and give an asymptotic formula of T n. As applications, we show the asymptotic normality of many famous combinatorial numbers, such as the Stirling numbers of the second kind, the Whitney numbers of the second kind, the r-Stirling numbers and the r-Whitney numbers of the second kind.