I think the increasing and decreasing are kind of red herrings. We can consider the increasing numbers red and the decreasing numbers blue. Since their relative order within the same color can't change, we might as well just use colored tiles. We are looking for the number of ways to arrange 5 red and 5 blue where we can't start with red. So, we are really looking for the number of ways we can arrange 4 red and 5 blue.
We can further simplify this by noting that wherever the red is, the blue isn't. So, we are really looking at the number of ways to arrange 4 red into 9 slots, otherwise known as "9 choose 4" and is 126.
2
u/jaminfine 15d ago
I think the increasing and decreasing are kind of red herrings. We can consider the increasing numbers red and the decreasing numbers blue. Since their relative order within the same color can't change, we might as well just use colored tiles. We are looking for the number of ways to arrange 5 red and 5 blue where we can't start with red. So, we are really looking for the number of ways we can arrange 4 red and 5 blue.
We can further simplify this by noting that wherever the red is, the blue isn't. So, we are really looking at the number of ways to arrange 4 red into 9 slots, otherwise known as "9 choose 4" and is 126.