How many automobiles licence plates can be made if the inscriptions on each contains two different letters followed by three differents digits ?

Answer

To find the total number of automobile license plates that can be made with the given conditions, we will follow these steps:

### Step 1: Determine the number of ways to choose the letters
We need to select 2 different letters from the 26 letters of the English alphabet. The first letter can be any of the 26 letters, and the second letter must be different from the first.

- For the first letter: 26 choices
- For the second letter: 25 choices (since it must be different from the first)

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

### Step 2: Determine the number of ways to choose the digits
Next, we need to select 3 different digits from the 10 digits (0-9). The first digit can be any of the 10 digits, the second digit must be different from the first, and the third digit must be different from both the first and the second.

- For the first digit: 10 choices
- For the second digit: 9 choices (since it must be different from the first)
- For the third digit: 8 choices (since it must be different from the first two)

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

### Step 3: Calculate the total number of license plates
Now, we multiply the number of ways to choose the letters by the number of ways to choose the digits to find the total number of license plates.

\[ \text{Total License Plates} = (26 \times 25) \times (10 \times 9 \times 8) \]

### Step 4: Perform the calculations
Calculating the values:
1. Calculate the letters:
\[ 26 \times 25 = 650 \]
2. Calculate the digits:
\[ 10 \times 9 \times 8 = 720 \]
3. Now multiply the results:
\[ 650 \times 720 = 468000 \]

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