How to count the number of subsets of $X$ of size $k$ that are disjoint from $A$?

Question

Apologies if my formatting is incorrect, this is my first post.

I'm currently taking Discrete Mathematics, but I'm struggling to understand more complicated uses of PIE. The only examples we've covered in class are things such as giving out cookies using binomial coefficients, but we've never done anything like this problem, although this problem is labeled as 1.6 in our book, we've got no examples similar, unless I'm not interpreting the problem correctly. Any help?

Let $X = \{1, 2, 3, \dots, n\}$ and $A = \{1, 2, 3, \dots, k\}$ for some $n \ge k \ge 0$.

We say that a subset $B$ of $X$ is disjoint from $A$ if $A \cap B = \emptyset$. Using PIE, count the number of subsets of $X$ of size $k$ that are disjoint from $A$.

Answer 1

For PIE, the $k$ properties to be avoided here are $i\in B$, where $i=1$ to $k$. For a given count of $r$ properties, there are $\binom{k}{r}$ choices of properties and then $\binom{n-r}{k-r}$ ways to complete the $k$-subset, so PIE yields $$\sum_{r=0}^k (-1)^r \binom{k}{r} \binom{n-r}{k-r} =\binom{n}{k}-k \binom{n-1}{k-1}+\binom{k}{2} \binom{n-2}{k-2}-\binom{k}{3} \binom{n-3}{k-3}+\dots+(-1)^k \binom{n-k}{0}$$

Answer 2

Probably breaking protocol here, but I'm a noob so don't have enough reputation to respond to RobPratt with a comment. I'm trying to help my nephew with discrete mathematics, and he got this same question. We got here by trial & error, but I think Rob's answer is off a bit. This is what we had to use:

$$\binom{n}{k} - \binom{k}{1}\binom{n-k}{k-1} - \binom{k}{2}\binom{n-k}{k-2} - \binom{k}{3}\binom{n-k}{k-3} - \dots - \binom{k}{k}\binom{n-k}{k-k}$$

I wanted to share in case this is the correct answer, and if by chance Rob might see this and comment.