To find the number of words formed by permuting all the letters of the word INDEPENDENCE such that the E's do not come together, we can use the principle of inclusion-exclusion.

Step 1: Total number of permutations
The total number of permutations of the letters in the word INDEPENDENCE is given by the formula:
Total permutations = n! / (n1! * n2! * n3! * ... * nk!)
where n is the total number of letters and n1, n2, n3, ..., nk are the frequencies of each distinct letter.

In this case, the word INDEPENDENCE has a total of 13 letters, with the following frequencies:
- I: 1
- N: 3
- D: 2
- E: 4
- P: 1
- C: 1

Using the formula, we can calculate the total number of permutations as:
Total permutations = 13! / (1! * 3! * 2! * 4! * 1! * 1!) = 13! / (1 * 6 * 2 * 24 * 1 * 1) = 13! / 288

Step 2: Counting the number of permutations where the E's come together
To count the number of permutations where the E's come together, we treat the two E's as a single unit. This reduces the problem to finding the number of permutations of the following letters: INDNPDNCE.

Now, the total number of letters is reduced to 9, with the following frequencies:
- I: 1
- N: 2
- D: 2
- P: 1
- C: 1

Using the formula, we can calculate the number of permutations where the E's come together as:
Permutations with E's together = 9! / (1! * 2! * 2! * 1! * 1!) = 9! / (1 * 2 * 2 * 1 * 1) = 9! / 4

Step 3: Counting the number of permutations where the E's do not come together
To find the number of permutations where the E's do not come together, we subtract the number of permutations where the E's come together from the total number of permutations.

Number of permutations without E's together = Total permutations - Permutations with E's together
= 13! / 288 - 9! / 4

Simplifying this expression, we get:
Number of permutations without E's together = (13!/288) - (9!/4)
= (13 * 12 * 11 * 10 * 9! / 288) - (9! / 4)
= 9! * (13 * 12 * 11 * 10 / 288 - 1 / 4)
= 9! * (11 * 10 - 9) / 4
= 9! * (110 - 9) / 4
= 9! * 101 / 4

Step 4: Calculating the final answer
Using the formula for n!, we can calculate 9! as:
9! = 9 * 8 * 7 * 6 * 5 * 4 *