In the previous question (no.46), how many different plates can be formed if the repetitions are not allowed?

Options

A. A) 468000
B. B) 13000
C. C) 15680
D. D) none of these

Answer

To solve the problem of how many different plates can be formed if repetitions are not allowed, we will follow these steps:

### Step 1: Determine the number of choices for letters
We need to select 2 letters from the English alphabet. Since there are 26 letters in total and repetitions are not allowed, we can use the permutation formula:

\[ \text{Number of ways to choose 2 letters} = P(26, 2) = \frac{26!}{(26 - 2)!} = \frac{26!}{24!} = 26 \times 25 = 650 \]

### Step 2: Determine the number of choices for digits
Next, we need to select 3 digits from the 10 possible digits (0 through 9). Again, since repetitions are not allowed, we use the permutation formula:

\[ \text{Number of ways to choose 3 digits} = P(10, 3) = \frac{10!}{(10 - 3)!} = \frac{10!}{7!} = 10 \times 9 \times 8 = 720 \]

### Step 3: Calculate the total number of different plates
Now, to find the total number of different plates that can be formed, we multiply the number of ways to choose the letters by the number of ways to choose the digits:

\[ \text{Total number of plates} = \text{Number of ways to choose letters} \times \text{Number of ways to choose digits} \]
\[ \text{Total number of plates} = 650 \times 720 = 468000 \]

### Conclusion
Thus, the total number of different plates that can be formed if repetitions are not allowed is **468000**.

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