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-ball.

Question

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-ball.
For example, if a 1-ball and two 2-balls are rolled down a 3-trough in the sequence 1, 2, 2, then each ball will drop through a hole. However, not all balls in the sequence 3, 1, 3 will drop. A sequence of balls is called a sinkable sequence if each ball drops through a hole.
a Explain why the sequence 3, 1, 4, 3 is not sinkable on a 4-trough.
b Explain why every sequence of five balls with diameters 1, 2, 3, 3, 4 is
sinkable on a 5-trough.
c List all sinkable sequences for a 3-trough.
d Howmanysinkablesequencesfora5-troughcontainexactlyone4-ball and exactly one 5-ball?