We can’t use the most basic approach of counting how many ways there are to place the first ball, and so on, because there is no “first ball” as far as the result is concerned. The balls are all alike (“indistinguishable”), so we don’t know or care which is in which basket but we do care how many balls are in basket 1, how many in basket 2, and so on. Picture, say, 3 baskets in a row, and 5 balls to be put in them. “Balls in urns” are a classic way to illustrate problems of this type today, I rarely see the word “urn” outside of combinatorics, and more often use words like “boxes” or “bags” or “bins”. Where the notation C(n,r) means the number of ways to choose r things from n things, i.e., the usual binomial coefficient. The number of different ways we can place b indistinguishable balls into u distinguishable urns is: I hope someone could please help me to prove this result: We’ll start with a simple example from 2001 that introduces the method: Placing Balls in Urns (I only remember the method, not the formulas.) Balls in urns Today, we’ll consider a special model called Stars and Bars, which can be particularly useful in certain problems, and yields a couple useful formulas. We have been looking at ways to count possibilities (combinatorics), including a couple ways to model a problem using blanks to fill in.
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