# Binomial Distribution at Infinity

Consider a binomial distribution of $X\sim B(n,p)$.

It can be easily shown that $P(X=k)={n\choose k}p^k{(1-p)}^{n-k}$ for $k=0,1,2,3,\ldots,n$.

Now, let's take the limit of the above using $n \to \infty$. Instead of having an infinitesimal $p$, let's assume that it is given that $np$, the mean of the probability distribution function, is some finite value $m$.

Find $P(X=k)$ in terms of $m$ and $k$ for this new distribution, where $k=0,1,2,3,\ldots$, without looking anything up or reciting any formulas from memory.

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