Show that a cube Qn includes a number ofQk cubes.
(We know that k<=n.)
Please define a cube Qn or Qk. Is that the number of dimensions of the cube? I am picturing vertices corresponding to n-tuples where each coordinate is in $\{0,1\}$ and edges between vertices if and only if the vertices differ by exactly one coordinate.
Ok, so then you simply need to describe the way to count the number of Qk cubes. Because the set of $2^n$ will include differences that are fixed on the $n-k$ coordinates that we are not considering as the subcube and change on exactly one coordinate among the remaining $k$ coordinates, this is a straightforward application of the product principle.
So, it is the product where we choose the $k$ coordinates for the Qk subcube (there are $\dbinom{n}{k}$ ways to choose them). Independent of that choice, we have $n-k$ remaining coordinates, and those coordinates can be any fixed set of coordinates for a given Qk cube. There are $2^{n-k}$ ways to fix those coordinates. By the Product Principle, the total number of subcubes of dimension k will be $\dbinom{n}{k}2^{n-k}$