Solution:

The graph passes through $(1,25)$ exactly when $f(1 - a) = 25.$ Let $t = 1 - a,$ so we just need the number of solutions to $f(t) = 25.$ Factor $f(x) = 100x(x-1)(x-2),$ with roots $0, 1, 2.$ Its local maximum on $(0,1)$ is $f(0.5) = 37.5,$ which beats $25.$ So the line $y = 25$ cuts the cubic in $3$ points. Each one gives a single $a,$ so there are $3$ values. Thus, C is the correct answer.