How many automobile license plates can be made, if each plate contains two different letters followed by three different digits ?

Answer

To solve the problem of how many automobile license plates can be made with two different letters followed by three different digits, we will break it down step by step.

### Step 1: Determine the number of choices for letters
We need to select 2 different letters from the English alphabet, which has 26 letters.

- For the first letter, we have 26 options.
- For the second letter, since it must be different from the first, we have 25 options.

Thus, the total number of ways to choose the letters is:
\[ 26 \times 25 \]

### Step 2: Determine the number of choices for digits
Next, we need to select 3 different digits from the digits 0 to 9, which gives us a total of 10 digits.

- For the first digit, we have 10 options.
- For the second digit, since it must be different from the first, we have 9 options.
- For the third digit, since it must be different from both the first and second, we have 8 options.

Thus, the total number of ways to choose the digits is:
\[ 10 \times 9 \times 8 \]

### Step 3: Combine the choices
Now, we combine the choices for letters and digits. The total number of different license plates can be calculated by multiplying the number of ways to choose the letters by the number of ways to choose the digits:
\[ (26 \times 25) \times (10 \times 9 \times 8) \]

### Step 4: Calculate the total
Now we perform the calculations:
1. Calculate the letters:
\[ 26 \times 25 = 650 \]
2. Calculate the digits:
\[ 10 \times 9 \times 8 = 720 \]
3. Multiply the results:
\[ 650 \times 720 = 468000 \]

### Final Answer
The total number of different automobile license plates that can be made is:
\[ 468000 \]