  In the year 1990, B\'ela Bollob\'as, Imre Leader and Andrew Radcliffe
considered the following combinatorial problem: given three parameters k, n and
q, find a set of k vertices in the binary n-cube which contains a maximal
number of q-dimensional subcubes. It was shown that an optimal solution is
given by the k vertices which coincide with the binary representations of the
number 0 , 1 , ... , k-1. Two proofs were presented. The proof given by
Bollobas and Leader is particularly elegant and short. Here we show that also
the other proof, the one given by Bollobas and Radcliffe, becomes quite simple
and short when it is combined with a lemma from Graham whose publication dates
back to 1970. As a second application of Graham's lemma, we solve a recursive
equation (related to the optimization problem that we discussed before) that
might be considered interesting in its own right.
