To determine how many ways the letters of the word "REMEMBER" can be arranged such that neither the two 'R's nor the three 'E's are together, we can follow a step-by-step approach:

Step 1: Identify distinct letters

The word "REMEMBER" contains 8 letters, with the following distribution:

• R: 2 occurrences
• E: 3 occurrences
• M: 2 occurrences
• B: 1 occurrence

Step 2: Arrange the distinct non-repeated letters

First, we arrange the letters without considering the restrictions on the 'R's and 'E's. These letters are:

• M, M, B

Without restrictions, we can think of these as placeholders where the 'R's and 'E's will be inserted later. The 5 non-repeating letters (R, M, E, M, B) can be arranged in:

$\frac{5!}{2! \times 2!} = \frac{120}{4} = 30 \text{ ways}.$

Here, we divide by $2!$ for the repeated 'M's and $2!$ for the repeated 'R's.

Step 3: Insert the R's and E's

Now, we need to place the 'R's and 'E's such that no two 'R's or two 'E's are together. After placing the distinct letters, there are 6 possible gaps between the letters (one before each letter, between each pair, and one after the last letter).

We can visualize it as:

$\_ \text{M} \_ \text{M} \_ \text{B} \_$

Step 4: Place the R's in gaps

We need to place the 2 'R's in different gaps. The number of ways to do this is:

$\binom{6}{2} = 15 \text{ ways}.$

Step 5: Place the E's in gaps

Next, we place the 3 'E's in the remaining 4 gaps. Since we need to ensure that no two 'E's are together, we can choose 3 distinct gaps out of the remaining 4. The number of ways to do this is:

$\binom{4}{3} = 4 \text{ ways}.$

Step 6: Total number of arrangements

Now, we multiply the results from each step to find the total number of valid arrangements:

$30 \times 15 \times 4 = 1800.$

Thus, the total number of ways to arrange the letters of "REMEMBER" such that no two 'R's or 'E's are together is 1,800.

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Would you like more details or have any questions? Here are 5 related questions to explore further:

1. How would the arrangement change if there were 3 R's instead of 2?
2. How can you calculate arrangements if only the R's must be separate, but E's can be together?
3. What would the formula be if all letters in the word were distinct?
4. How do the constraints change if we also ensure that no M's are adjacent?
5. How many ways are there to arrange the word REMEMBER without any restrictions?

Tip: For problems involving restrictions on letter arrangements, always arrange the unrestricted elements first and then place the restricted ones carefully.