Question 3
Let $n$ be a positive integer and let $k$ be a positive integer not exceeding $n$. Determine the number of ordered $k$-element tuples $(a_1,a_2,\ldots,a_n)$ satisfying:
(a) All $a_i$ are distinct and belong to $\{1,2,\ldots,n\}$.
(b) $a_r\gt a_s$ for some $r\lt s$.
(c) $a_r$ has the same parity as $r$ for some $r$.
(a) All $a_i$ are distinct and belong to $\{1,2,\ldots,n\}$.
(b) $a_r\gt a_s$ for some $r\lt s$.
(c) $a_r$ has the same parity as $r$ for some $r$.