How many different permutations of the letters A, B, C, D, E and F are there if each letter is used only once. How many of these end in ED?

To solve this problem, we can break it down into two parts:

### 1. **Total number of permutations of the letters A, B, C, D, E, F:**

The total number of permutations of \(n\) distinct objects is given by \(n!\). 

Here, \(n = 6\) (since we have 6 letters: A, B, C, D, E, F).

\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\]

So, there are **720 different permutations** of the letters A, B, C, D, E, F.

### 2. **Number of permutations that end in "ED":**

If the permutation must end in "ED," then we are left with the letters A, B, C, F to arrange in the first four positions.

The number of permutations of the remaining 4 letters is given by \(4!\).

\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]

So, there are **24 permutations** of the letters A, B, C, D, E, F that end in "ED."

### Summary:
- **Total number of permutations:** 720
- **Number of permutations that end in "ED":** 24

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How many permutations of the letters A, B, C, D, E, and F start with "A"?
2. How many permutations of the letters A, B, C, D, E, and F have "AB" together?
3. If "A" and "B" must be at the beginning and end, how many permutations are there?
4. What is the number of permutations of the letters A, B, C, D, E, and F that have "EF" together?
5. How many permutations of the letters A, B, C, D, E, and F have no two vowels together?

**Tip:** When calculating permutations, fixing positions of certain elements first simplifies the problem.

Explore the number of different permutations of the letters A, B, C, D, E, and F, focusing on those that end specifically in "ED". Learn how permutations are calculated using factorial formulas. Suitable for students in Grades 7-9.

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