For a Steiner system, how many blocks intersect exactly one position of a specific block? Consider Steiner system $S(2,k,v)$ with $2 = t < k < v$, a family of $k$-subsets of finite set $S$ with $|S|=v$ such that each $t$-subset of $S$ is contained in exactly one block.
A paper I'm reading states without proof that, for some specific block $B$, "the number of blocks meeting $B$ in exactly one specified point" (containing exactly one specified element of $B$) is $\frac{v-k}{k-1}$, and therefore the number meeting $B$ in a single point is $\frac{k(v-k)}{k-1}$.
It also says the number of blocks not meeting $B$ at all is $\chi=\frac{v(v-1)}{k(k-1)}-\frac{k(v-k)}{k-1}-1$.
Where did these expressions come from?
 A: Since $t=2$, any block $B'\neq B$ is either disjoint from $B$, or meets $B$ in exactly one point $x\in B\cap B'$.
Now how many blocks $B'$ are there that meet $B$ in exactly one specified point $x$? Every such block $B'$ must also have at least one point $z \in S\setminus B$.
You can count all such points $z$ (there are $|S\setminus B| = v - k$) of them, but since different values of $z$ might yield the same block $B'$, you will have to divide that number by the size of $B'\cap (S\setminus B) = B'\setminus B$ (which is $k-1$ by the initial remark).
In summary, the number of blocks $B'$ meeting $B$ in $x$ is $\frac{v-k}{k-1}$, and since $B$ contains $k$ such points $x$, multiplying that fraction by $k$ gives you the number of blocks $B'$ meeting $B$ in some single point.
Your last equation is simply a way to re-phrase the initial remark: The number of blocks disjoint from $B$ is the total number of blocks ($\frac{v(v-1)}{k(k-1)}$) minus the number of blocks intersecting $B$ in one point (the number just calculated) minus 1 (for the block $B$ itself).
