The number of different words that can be formed with the letters of the word ORDINATE so that
Beginning with O is….

Answer

To solve the problem of finding the number of different words that can be formed with the letters of the word "ORDINATE" that begin with the letter 'O', we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Letters**: The word "ORDINATE" consists of 8 letters: O, R, D, I, N, A, T, E.

2. **Fix the First Letter**: Since we want the words to begin with 'O', we fix 'O' as the first letter. This leaves us with the remaining letters: R, D, I, N, A, T, E.

3. **Count the Remaining Letters**: After fixing 'O', we have 7 letters left: R, D, I, N, A, T, E.

4. **Calculate the Number of Arrangements**: The number of different arrangements of these 7 letters can be calculated using the factorial of the number of letters. Since all the remaining letters are distinct, the number of arrangements is given by 7! (7 factorial).

5. **Compute 7!**:
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

6. **Conclusion**: Therefore, the total number of different words that can be formed with the letters of the word "ORDINATE" beginning with 'O' is 5040.

### Final Answer:
The number of different words that can be formed with the letters of the word "ORDINATE" that begin with 'O' is **5040**.