Distribution of $n$ identical balls and $n$ distinct balls to $m$ children or boxes

Question

I saw in my book that number of ways of distributing $n$ distinct objects in $m$ boxes is $m^n$, but when I got a question like the number of ways of distributing $15$ identical toys to $6$ children such that each one gets at least one toy is $14C5$. How is this? Please explain.

Answer 1

Imagine the 15 toys in a line and place 5 vertical lines in 5 gaps chosen from the 14 gaps between adjacent toys, so you are choosing 5 from a set of 14. The toys up to first bar go to Alice, the next bunch go to Bob, etc. Convince yourself that each possible toy assignment corresponds to exactly one "bar insertion choice." This also counts the number of ways to write 15 as an ordered sum of 6 positive integers, which should be easy to see. The technique has become known as "Stars-and-Bars."