How many three letter words can be formed using only consonants but only once ?

Answer

To solve the problem of how many three-letter words can be formed using only consonants without repetition, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Consonants**:
The English alphabet consists of 26 letters, out of which 5 are vowels (A, E, I, O, U). Therefore, the total number of consonants is:
\[ 26 - 5 = 21 \]

2. **Determine the Number of Choices for Each Letter**:
- For the **first letter**, we can choose any of the 21 consonants.
- For the **second letter**, since we cannot repeat the first letter, we have 20 consonants left to choose from.
- For the **third letter**, we cannot repeat either of the first two letters, leaving us with 19 consonants to choose from.

3. **Calculate the Total Combinations**:
The total number of different three-letter combinations can be calculated by multiplying the number of choices for each letter:
\[ \text{Total Combinations} = 21 \times 20 \times 19 \]

4. **Perform the Multiplication**:
Now, we calculate:
\[ 21 \times 20 = 420 \]
\[ 420 \times 19 = 7,980 \]

5. **Final Answer**:
Therefore, the total number of three-letter words that can be formed using only consonants without repetition is:
\[ \boxed{7,980} \]