Question 4
Let $(a_1, a_2, a_3, \ldots, a_{12})$ be a permutation of $(1, 2, 3, \ldots, 12)$ for which $$
a_1 \gt a_2 \gt a_3 \gt a_4 \gt a_5 \gt a_6
$$ and $$
\begin{aligned} &a_6 \lt a_7 \lt a_8 \lt a_9 \\ &\lt a_{10} \lt a_{11} \lt a_{12}. \end{aligned}
$$ An example of such a permutation is $(6, 5, 4, 3, 2, 1, 7, 8, 9, 10, 11, 12).$ Find the number of such permutations.
a_1 \gt a_2 \gt a_3 \gt a_4 \gt a_5 \gt a_6
$$ and $$
\begin{aligned} &a_6 \lt a_7 \lt a_8 \lt a_9 \\ &\lt a_{10} \lt a_{11} \lt a_{12}. \end{aligned}
$$ An example of such a permutation is $(6, 5, 4, 3, 2, 1, 7, 8, 9, 10, 11, 12).$ Find the number of such permutations.