ILL 1983 Problem 17

In how many ways can $1, 2,\ldots, 2n$ be arranged in a $2 \times n$ rectangular array $\left(\begin{array}{cccc}a_1& a_2 & \cdots & a_nb_1& b_2 & \cdots & b_n\end{array}\right)$ for which:(i) $a_1 < a_2 < \cdots < a_n,$ (ii) $b_1 < b_2 <\cdots < b_n,$ (iii) $a_1 < b_1, a_2 < b_2, \ldots, a_n < b_n \ ?$