= Solution
Suppose $y$ in the <capped simplex> has a coordinate strictly between zero and one. Since $\sum_i y_i=k$ is an integer, it cannot have exactly one such coordinate: the other coordinates would contribute an integer sum. Hence there are distinct $i,j$ with $0<y_i,y_j<1$.
Choose $0<\varepsilon<\min\{y_i,1-y_i,y_j,1-y_j\}$ and set $y^{\pm}=y\pm\varepsilon(e_i-e_j)$, where $e_i,e_j$ are <standard basis> vectors. Both perturbed <vectors> satisfy the coordinate bounds and have the same coordinate sum. They are distinct and $y=(y^++y^-)/2$, so $y$ is not an <extreme point>.
Conversely, every zero-one <vector> in this <convex polytope> is an <extreme point>. If it were a nontrivial <convex combination> of two feasible <vectors>, each coordinate equal to zero would force both corresponding coordinates to be zero, and each coordinate equal to one would force both to be one. Thus the two <vectors> would equal the original one. The <extreme points> are therefore exactly the indicators of $k$-element subsets.
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