Number of subsets not containing a given subset and a given point... 1) Let $V$ a set of cardinality $n$. Let $B$ a subset of cardinality $2 \le k<n$, $v \in B$ and $l$ a number such that $k \le l \le n-2 $. How many $l-$subsets not containing neither $B$ nor $v$ are there?
2) Let $V$, $B$ and $v$ as before. Let $l$ a number such that $k \le l \le n-1$. How many $l-$subsets not containing $B$ but containing $v$ are there?
 A: In the first problem it’s easier to count the complementary sets. There are exactly as many $\ell$-subsets of $V$ not containing $B$ or $v$ as there are $(n-\ell)$-subsets of $V$ that contain $v$ and at least one element of $B$. To form an $(n-\ell)$-subset of $V$ that contains $v$ and at least one element of $B$, we start by choosing $v$. Then we must choose $n-\ell-1$ elements of $V\setminus\{v\}$, which we can do in $\binom{n-1}{n-\ell-1}$ ways. However, $\binom{n-k-1}{n-\ell-1}$ of these subsets are disjoint from $B$, so we can’t use them. That leaves a total of $$\binom{n-1}{n-\ell-1}-\binom{n-k-1}{n-\ell-1}=\binom{n-1}\ell-\binom{n-k-1}{\ell-k}$$ subsets of the desired form.
In the second problem it is again easier to count the complementary sets; they are the $(n-\ell)$-subsets of $V$ that do not contain $v$ but do contain at least one element of $B$. There are $\binom{n-1}{n-\ell}$ $(n-\ell)$-subsets of $V$ that do not contain $v$, but $\binom{n-k-1}{n-\ell}$ of them are disjoint from $B$; that leaves a total of $$\binom{n-1}{n-\ell}-\binom{n-k-1}{n-\ell}=\binom{n-1}{\ell-1}-\binom{n-k-1}{\ell-k-1}$$ sets of the desired form.
