What is the expected number of bins that ends not being empty?

Question

Consider a situation in which 8 distinct balls are distributed among 6 distinct bins. What is the expected number of bins that ends not being empty ?

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Can I apply Sterling formula of 2nd kind here ?

Answer 1

Let $X_i$ be the indicator variable for the $i^{th}$ bin. Thus $X_i=1$ if that bin contains a ball, and $0$ otherwise. By Linearity of Expectation, the answer you seek is $$E=\sum_{i=1}^6E[X_i]$$ Of course, $E[X_i]$ is just the probability that the $i^{th}$ bin is non-empty, hence $E[X_i]=1-\left(\frac 56\right)^8$. Thus $$E=6\times \left[1-\left(\frac 56\right)^8\right]=4.604591764$$