A trough has holes with diameter 1, 2, 3, ... , n along its base in that order. It is called an n-trough. The trough is mounted on legs so it is tilted with the smallest hole uppermost. With each n-trough there are n balls, each with integer diameter at most n. Two or more balls may have the same diameter. They are rolled down the trough one at a time, not necessarily in size order. If a ball drops through a hole, it triggers a trapdoor that closes the hole. A ball with diameter d is called a d hole.
A sequence of balls where all balls sink into a hole is called a sinkable sequence. For example, in a 3 trough the sequence 1, 2, 2 is sinkable. But 3, 1, 3 is not.
a) How many sinkable sequences for a 5-trough contain exactly one 4-ball and exactly one 5-ball?
I have worked my way through the different combinations ([1,2,3,4,5], [1,1,1,4,5], etc.) and wound up with 320 different sequences. Am I right? Could someone please help me check my working as my combinatorics is not the best....