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consider the task of dividing 25 observations into 5 groups.

My question is how to get that there are $2.4\times10^{15}$ different ways to arrange those observations into 5 groups.

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The number of ways to partition a set of $n$ objects into $k$ non-empty subsets is given by the Stirling number of the second kind $S(n,k)$ also noted $\textstyle \lbrace {n \atop k}\rbrace$.

$$\textstyle \lbrace {25 \atop 5}\rbrace= 2436684974110751=2.44\times 10^{15}$$ I suggest you look at the following paper for the demonstration
http://www.elcamino.edu/faculty/gfry/210/DistributeDifBallsDifBoxes.pdf

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