There are five gentlemen and four ladies to dine at a round table. In how many ways can they
themselves, so that no two ladies are together
To ensure that no two ladies are together, we can place the gentlemen first in the clockwise direction, leaving gaps for the ladies to be placed in between them.
We can think of this as placing the gentlemen at the 5 vertices of a pentagon, and then placing the ladies in the gaps between the gentlemen.
We can fix one gentleman at a vertex and think of the other gentlemen as circular permutations around the pentagon. There are 4 gentlemen left to be placed, so there are 4! = 24 ways to arrange them.
Now, we need to place the ladies in the gaps between the gentlemen. There are 4 gaps between the 5 gentlemen, so we can think of this as placing 4 ladies in the 4 gaps.
We can think of this as distributing 4 identical objects (the ladies) into 4 distinct boxes (the gaps). This can be done in 4^4 = 256 ways using the stars and bars method.
Therefore, the total number of ways to arrange the gentlemen and ladies is 24 * 256 = 6,144 ways.