The number of value of k for which the equation x³ – 3x + k = 0 has two different roots lying in the interval (0, 1) are

no value of k satisfies the requirement

Let f(x) = x³ – 3x + k

f'(x) = 3(x – 1) (x + 1)

Thus the stationary points being $+1$ and $-1$ they lie outside the given region hence no value of k can make the equation roots lie in the given range of $(0,1)$.