Balls & Urns, Sticks & Stones, Stars & Bars

In how many ways can $n$ indistinguishable balls be placed into $k$ distinguishable boxes? What if each box must receive at least one ball? It turns out that the answers can be expressed as binomial coefficients. However, the standard journey to finding them requires an ingenious trick. We discuss and apply the technique in this video to derive the formula for counting compositions.