How many four-digit personal identification nhumbers are possible if no digit can be used twice and no number can begin with 0?

Options

A. 210
B. 4536
C. 6561
D. 10000

Explanation

To solve the problem of how many four-digit personal identification numbers are possible under the given conditions, we will follow these steps:

### Step 1: Determine the options for the first digit
The first digit of the four-digit number cannot be 0 (as per the problem statement). Therefore, the possible digits for the first position are from 1 to 9. This gives us 9 options.

**Hint:** Remember that the first digit cannot be zero; it must be a non-zero digit.

### Step 2: Determine the options for the second digit
For the second digit, we can use any digit from 0 to 9, but we cannot use the digit that has already been used in the first position. Since we have already used one digit, we have 9 remaining options for the second digit.

**Hint:** The second digit can include 0, but not the digit chosen for the first position.

### Step 3: Determine the options for the third digit
For the third digit, we again can use any digit from 0 to 9, but we cannot use the digits that have already been used in the first and second positions. Therefore, we have 8 remaining options for the third digit.

**Hint:** The third digit must be different from both the first and second digits.

### Step 4: Determine the options for the fourth digit
For the fourth digit, we again can use any digit from 0 to 9, but we cannot use the digits that have already been used in the first, second, and third positions. This leaves us with 7 remaining options for the fourth digit.

**Hint:** The fourth digit must be different from the first three digits.

### Step 5: Calculate the total number of combinations
Now, we can calculate the total number of four-digit personal identification numbers by multiplying the number of options for each digit:

\[ \text{Total combinations} = (\text{options for 1st digit}) \times (\text{options for 2nd digit}) \times (\text{options for 3rd digit}) \times (\text{options for 4th digit}) \]

Substituting the values we found:
\[ \text{Total combinations} = 9 \times 9 \times 8 \times 7 \]

Calculating this gives:
\[ 9 \times 9 = 81 \]
\[ 81 \times 8 = 648 \]
\[ 648 \times 7 = 4536 \]

Thus, the total number of four-digit personal identification numbers possible is **4536**.

### Final Answer
The answer is **4536**.

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