I am totally guessing at this one.

Let k and n be positive integers satisfying k < n. How many subsets of [n] are not also subsets of [k]?

Take k=2 and n=3, then

[3] = {$\displaystyle \phi$} {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

[2] = {$\displaystyle \phi$} {1} {2} {1,2}

The cardinality of [3] = $\displaystyle 2^3$ and the cardinality of [2] = $\displaystyle 2^2$. The number of subsets of [3] that are not subsets of [2] = $\displaystyle 2^3$ - $\displaystyle 2^2$.

So I think the number of subsets of [n] that are not also subsets of [k] would be $\displaystyle 2^n$ - $\displaystyle 2^k$. Is this correct?