446. The "Effervescent" Puzzle. In how many ways can the letters in the word EFFERVESCES \({ }^{*}\) be arranged in a line so that no two \(\mathrm{E}\) are adjacent? Of course, we do not distinguish between identical letters like \(F F\), since swapping them does not result in a new arrangement.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-155.jpg?height=468&width=462&top_left_y=1431&top_left_x=800)

When the reader has figured this out, they can try to find the answer under the same conditions when the letters are arranged in a circle (see the figure). Of course, we are interested in the order of the letters, not their position on the circle; moreover, reading should always be clockwise, as shown in the figure.
446. In the first case, there are 88200 ways. There is one simple method by which the answer can be obtained, but explaining it would require too much space. In the second case, the answer is reduced to 6300 ways.