This is a statistics problem. although this is not a problem which needs an answer, I want to know the reason Why this is right. Can you guys help me?
A deck of $n=10$ cards is numbered from 1 to 10. The cards are shuffled and laid down from left to right, face up. Order each of the five successive pairs of cards. Each of these five pairs determines a random interval. Let $X$ equal the number of these five random intervals that intersect each of the other four intervals. It can be shown that :
$$\begin{align} P(X\geq k)=\frac{2^k}{2k+1\choose k}\end{align}$$