Let $M=(E,I)$ be a paving matroid with rank $n$. Let $A\subset E$ be an $n-1$ subset. How many bases of $M$ containing $A$ exist? (Note that every $n-1$ subset of $E$ is independent.)



Counting certain bases of a paving matroids
In an $n$-dimensionsl space, take a hyperplane $H$. Take a set of vectors in $H$ in a general position, and add one vector $x$ outside $H$. This is a paving matroid of rank $n$.


Now, if $|A|=n-1$ and $x\notin A$, then the only vector which can be added without breaking independency is $x$, so the minimal number of sought bases is $1$.


If $x\in A$, then any vector can be added in order to form a base So the maximal number is $|E|-n+1$.


