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

Explanation

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

### Step 1: Choosing and Arranging the Letters
1. **Choose 2 different letters from 26 letters**:
The number of ways to choose 2 letters from 26 is given by the combination formula \( C(n, r) \), where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose.
\[ C(26, 2) = \frac{26!}{2!(26-2)!} = \frac{26 \times 25}{2 \times 1} = 325 \]

2. **Arrange the 2 letters**:
Since the order of the letters matters, we need to multiply by the number of arrangements of 2 letters, which is \( 2! \):
\[ 2! = 2 \]
Therefore, the total arrangements for the letters is:
\[ 325 \times 2 = 650 \]

### Step 2: Choosing and Arranging the Digits
1. **Choose 3 different digits from 10 digits (0-9)**:
The number of ways to choose 3 digits from 10 is given by:
\[ C(10, 3) = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

2. **Arrange the 3 digits**:
Since the order of the digits also matters, we multiply by the number of arrangements of 3 digits, which is \( 3! \):
\[ 3! = 6 \]
Therefore, the total arrangements for the digits is:
\[ 120 \times 6 = 720 \]

### Step 3: Total License Plates
Now, we combine the total arrangements of letters and digits to find the total number of license plates:
\[ \text{Total License Plates} = (\text{Arrangements of Letters}) \times (\text{Arrangements of Digits}) = 650 \times 720 \]

Calculating this gives:
\[ 650 \times 720 = 468000 \]

### Final Answer
Thus, the total number of automobile license plates that can be made is **468,000**.

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