Probability that a set of $n(n+1)/2$ elements will contain $1... n$ elements, respectively, of $n$ possibilities We opened a 'fun size' bag of Skittles this afternoon, and it contained 5 yellow, 4 red, 3 blue, 2 green, and 1 purple Skittle. If the Skittles only come in these 5 colors, they are chosen randomly from a vat with an infinite number of each color, and each bag contains 15 Skittles, what are the chances that there will be one color with each cardinality between 1 and 5? (Which color has which cardinality is immaterial) Generalize for $n$ colors and $n(n+1)/2$ Skittles per bag. P.S. I just realized that Skittles also come in orange... let's pretend that they don't for the purposes of this puzzle.

The particular colour pattern you described has probability $$\binom{15}{1,2,3,4,5}\cdot \frac{1}{5^{15}}.$$ Here we are using the _Multinomial Distribution_ (please see Wikipedia).

For the probability of a $1,2,3,4,5$ distribution, multiply by $5!$, the number of permutations of the colours.