Ball-and-urn

The ball-and-urn technique, also known as stars-and-bars, sticks-and-stones, or dots-and-dividers, is a commonly used technique in combinatorics.

It is used to solve problems of the form: how many ways can one distribute $k$ indistinguishable objects into $n$ distinguishable bins? We can imagine this as finding the number of ways to drop $k$ balls into $n$ urns, or equivalently to arrange $k$ balls and $n-1$ dividers. For example, $****||$ $***|*|$ $*|**|*$ $...$ represent the ways to put $4$ objects in $3$ bins. The number of ways to do such is ${4+3-1 \choose 3-1} = {6 \choose 2} = 15$ .

More generally, the number of ways to put $n$ objects into $k$ bins is ${n+k-1 \choose k-1}$ .

The number of ways to put $n$ objects into $k$ bins, where each bin must have at least 1 object in it, is ${n-1 \choose k-1}$ .

Reasoning 1

Arranging $k$ *'s and $n-1$ |'s is the same as saying there are $k+n-1$ positions: $\underbrace{\_~~ \_~~ \_~~ \_~~ \_~~ \_~~}_{k+n-1}$ and you want to fill $k$ of them with *'s and the rest of them with |'s. Thus you are choosing $k$ positions out of $n+k-1$ total positions, resulting in a total of ${n+k-1\choose n-1}$ ways.

Reasoning 2

If you could only put one ball in each urn, then there would be ${n \choose k}$ possibilities; the problem is that you can repeat urns, so this does not work. You can, however, reframe the problem as so: imagine that you have the $n$ urns (numbered 1 through $n$ ) and then you also have $k-1$ urns labeled "repeat 1st", "repeat 2nd", ..., and "repeat $k-1$ -th". After the balls are in urns you can imagine that any balls in the "repeat" urns are moved on top of the correct balls in the first $n$ urns, moving from left to right. There is a one-to-one correspondence between the non-repeating arrangements in these new urns and the repeats-allowed arrangements in the original urns.

For a simple example, consider $4$ balls and $5$ urns. The one to one correspondence between several of the possibilities and the "repeated urns" version is shown. Since there are 4 balls, these examples will have three possible "repeat" urns. For simplicity, I am listing the numbers of the urns with balls in them, so "1,1,2,4" means $2$ balls in urn $1, 1$ in urn $2,$ and $1$ in urn $4.$ The same is true for the "repeat" urns options but I use the notation $r_1$ etc.

$1,2,3,4 \leftrightarrow 1,2,3,4$ (no repeats).
$1,1,2,4 \leftrightarrow 1,2,4,$ $r_1$ .
$1,2,2,2 \leftrightarrow 1,2,$ $r_2$ , $r_3$ .
$4,4,5,5 \leftrightarrow 4,5,$ $r_1$ , $r_2$ .

Since the re-framed version of the problem has $n+k-1$ urns, and $k$ balls that can each only go in one urn, the number of possible scenarios is simply ${n+k-1 \choose k}.$ Note: Due to the principle that ${a \choose b}={a \choose a-b}$ , we can say that ${n+k-1 \choose k}={n+k-1 \choose n-1}$ .

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Ball-and-urn

Reasoning 1

Reasoning 2

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