Intuition in Equivalence of ${n \choose r}$ and Permuting $n$ Objects Potentially dumb question warning. We have $$ \binom{n}{r} = \frac{n!}{r!(n-r)!} $$ which is viewed as the ways of picking $r$ objects from a group of $n$. It also happens to be the number of ways of permuting $n$ objects consisting of $r$ indistinct red objects and $n-r$ indistinct blue objects. I was wondering if someone could come up with a good intuition to link the two interpretations? Like an explanatory "bijection" between the permutations and the selections.

Look at it as choosing positions - the $r$ (or $n-r$) objects are choosing from $n$ possible positions, and each possible assignation, regardless of order, represents a permutation distinct with regard to the objects' respective sets.