(Solved): BathILy A popular activity in Bath, England, is visiting the heated public baths. The two most popul ...

BathILy A popular activity in Bath, England, is visiting the heated public baths. The two most popular are the Cross Bath (shown above) and New Royal Bath. A group of otters want to go for a swim and use both Bath baths. If there are just two Bath otters bathing, there's only one way they can split up to swim (one at each Bath bath). List all the different ways three otters (named \( o, t \), and \( T \) ) can split up to swim in two Bath baths. Fill in the blanks: Prove that your formula for the number of ways for \( n \) otters to split up and swim in two Bath baths is correct. There are actually a lot of Bath baths. A group of \( n \) otters is swimming in \( k \) Bath baths. There are \( O(n, k) \) ways for the otters to split up and swim in the Bath baths. Explain why \( O(k, k)= \) Suppose the youngest otter is in a Bath bath by itself. How many ways are there for the \( n \) otters to split up to swim in \( k \) Bath baths in this case? Why? Now suppose the youngest otter does not want to be by itself in a Bath bath. How many ways are there for the \( n \) otters to split up to swim in \( k \) Bath baths in this case? Why?