I'm quite sure it was posted before - have you searched for it?

Megus wrote: I'm quite sure it was posted before - have you searched for it? I can't it

Choosing a subset of $X$ with $k-1$ elements, we know that it can be a subset of at most one of $F_k$. Since each $F_k$ has $k$ subsets with $k-1$ elements, we can deduce that $|F_k| \le \frac{1}{k}\binom{n}{k-1}$ Let $t=[\log_{2}{n}]+1$. The problem is trivial if $t<k$, so assume that $t \ge k$. Let us pick a $M_k$, a subset of $X$, with $t$ elements. Then, the probability that an element in $F_k$ is also in $M_k$ is $\frac{\binom{n-k}{t-k}}{\binom{n}{t}}$. Let $T$ be the the number of elements in $F_k$ that is also in $M_k$. We have that $E(T) = |F_k|\frac{\binom{n-k}{t-k}}{\binom{n}{t}} \le \frac{\binom{t}{k}}{n-k+1}$. Using $\binom{t}{k} \le \frac{2^t}{\sqrt{t}} \le \frac{2n}{\sqrt{t}}$, we can get that $E(T) < 1$ and then we are done. $\blacksquare$