Let $n \geq 1$ and $X$ and $Y$ are $2$ disjoint sets, each of size $n$ elements.
An ordered triple $(A,B,C)$ of sets is called cool, if $A$ is subset of $X$, $B$ is subset of $Y$, $C$ is subset of $B$, and size of A + size of B = n
1)for $0 \leq k \leq n$. Determine the number of cool triples $(A,B,C)$ for which size of $A = k$.
2)for $0 \leq k \leq n$. Determine the number of cool triples $(A,B,C)$ for which size of $C = k$.
in this question I was able figure out part $1$ where size of $A$ is $k$ and so size of $B$ will be $n-k$ and size of $C$ will be $2^{n-k}$ as it is a subset of $B$ so number of cool triples will be $\binom{n}{k}\binom{n}{n-k}2^{n-k}$.
I am not sure about how to find the size of $A$ and $B$ for the second part.
$B$ must contain $C$, so $B$ can be formed by combining $C$ with some subset of the $n-k$ remaining elements of $Y$. Then $A$ can be any subset of $X$ of size $n - |B|$.
If we split this into cases based on the size of $B$ (call it $m$) you will get a sum (over $m=k, \ldots, n$) of the product of two binomial coefficients. I am not sure how to simplify this sum and/or if there is a neater formula for it.