In how many ways can we select n objects from a collection of size 2n that consists of n distinct and n identical objects? In how many ways can we select $n$ objects from a collection of size $2n$ that consists of $n$ distinct and $n$ identical objects?
The answer is $2^n$ and I really don't see how they get this. Selecting $n$ distinct from $2n$ is $\binom{2n}{n}$.

Let $A=\{a_1,\ldots,a_n\}$, where the $a_k$ are mutually distinguishable, and let $S$ be $A$ together with $n$ indistinguishable objects. An $n$-element subset $X$ of $S$ is completely determined when you know $X\cap A$: if $|X\cap A|=k$, the remainder of $X$ is just $n-k$ of the indistinguishable objects. $A$ has $2^n$ subsets, so there are exactly $2^n$ different possibilities for $X\cap A$ and therefore for distinguishable sets $X\subseteq S$ of cardinality $n$.
