Question 21
There are exactly two ways to insert the numbers 1, 2 and 3 into the circles $\circ\lt \circ\gt \circ$ such that every order relation $\lt $ or $\gt $ between numbers in adjacent circles is satisfied. The two ways are $1\lt 3\gt 2$ and $2\lt 3\gt 1$.
Find the total number of possible ways to insert the numbers 3, 14, 15, 9, 2 and 6 into the circles below, such that every order relation $\lt $ or $\gt $ between the numbers in adjacent pairs of circles is satisfied.
$$\circ\gt \circ\gt \circ\gt \circ\lt \circ\lt \circ.$$
Find the total number of possible ways to insert the numbers 3, 14, 15, 9, 2 and 6 into the circles below, such that every order relation $\lt $ or $\gt $ between the numbers in adjacent pairs of circles is satisfied.
$$\circ\gt \circ\gt \circ\gt \circ\lt \circ\lt \circ.$$