To solve this problem, we will address the constraints step by step, making sure both the conditions on the four $e$'s and the two $r$'s are satisfied.

Step 1: Treat the four $e$'s as a single block

Since the four $e$'s must be together, we can treat them as a single unit. This reduces the problem to arranging the remaining six letters: $r, r, g, n, a, t$, plus the block of four $e$'s. So, we are now working with 7 units (six letters plus the $e$-block).

Step 2: Place the two $r$'s with three letters in between

We are required to place the two $r$'s such that exactly three letters are between them. This reduces the problem to first placing the two $r$'s with the three-letter gap in one of the 7 positions available for these units.

To do this, consider the 7 available positions as follows:

$1, 2, 3, 4, 5, 6, 7$

We need to place one $r$ at position $i$ and the other $r$ at position $i+4$ so that exactly three positions remain between them. The possible positions for $i$ are $1, 2, 3$, since any larger value would exceed the 7 available positions.

Thus, we have 3 ways to place the $r$'s.

Step 3: Arrange the remaining letters

After placing the $r$'s, we need to arrange the remaining 5 units: $g, n, a, t$, and the block of $e$'s. There are 5! ways to arrange these remaining units. However, since there are identical letters (the two $e$'s), we must divide by the number of ways to arrange these identical letters, which is $2!$ for the repeated $e$'s.

Thus, the number of ways to arrange the remaining letters is:

$\frac{5!}{2!} = \frac{120}{2} = 60$

Step 4: Total number of arrangements

Now, multiply the number of ways to place the $r$'s (3 ways) by the number of ways to arrange the remaining letters (60 ways):

$3 \times 60 = 180$

Thus, the total number of different arrangements of the letters in "regenerate" where the four $e$'s are together and the two $r$'s have exactly three letters between them is 180.

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Would you like more details or clarifications on any part of the solution?

Here are 5 related questions you might find helpful:

1. How many arrangements of the letters in "regenerate" are there with no restrictions?
2. What if the two $r$'s must have exactly 2 letters between them?
3. How many arrangements of the letters in "regenerate" are there where no letters are adjacent?
4. What if all vowels (e, a) must be together in the word "regenerate"?
5. How would the solution change if the word had three $r$'s instead of two?

Tip: When dealing with identical objects in a permutation, always remember to divide by the factorial of the number of indistinguishable objects to avoid overcounting.