How many different words beginning with O and ending with E can be formed with the letters of the word ORDINATE, so that the words are beginning with O and ending with E

Options

A. `8!`
B. `6!`
C. `7!`
D. `7!2!`

Answer

To solve the problem of how many different words beginning with 'O' and ending with 'E' can be formed using the letters of the word "ORDINATE," we can follow these steps:

### Step 1: Identify the total letters in the word "ORDINATE"
The word "ORDINATE" consists of the following letters:
- O, R, D, I, N, A, T, E

Counting these letters, we find that there are a total of 8 letters.

### Step 2: Fix the positions of 'O' and 'E'
Since we want the words to begin with 'O' and end with 'E', we can fix these letters in their respective positions:
- First position: O
- Last position: E

### Step 3: Determine the letters to arrange
With 'O' and 'E' fixed, we are left with the following letters to arrange in the middle:
- R, D, I, N, A, T

This gives us a total of 6 letters (R, D, I, N, A, T) that can be arranged.

### Step 4: Calculate the number of arrangements
The number of different arrangements of these 6 letters can be calculated using the factorial notation. The number of ways to arrange n distinct objects is given by n! (n factorial).

Thus, for our 6 letters, the number of arrangements is:
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

### Conclusion
Therefore, the total number of different words that can be formed beginning with 'O' and ending with 'E' using the letters of "ORDINATE" is **720**.

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