How many natural numbers smaller than 10,000 are there in the decimal notation of which all the digits are different?

Options

A. 2682
B. 4474
C. 5274
D. 1448

Answer

To find how many natural numbers smaller than 10,000 have all different digits, we can break the problem down into counting 1-digit, 2-digit, 3-digit, and 4-digit numbers separately.

### Step 1: Count 4-digit numbers
A 4-digit number can be represented as \(abcd\), where \(a\), \(b\), \(c\), and \(d\) are the digits.

1. The first digit \(a\) can be any digit from 1 to 9 (9 options).
2. The second digit \(b\) can be any digit from 0 to 9 except for the digit chosen for \(a\) (9 options).
3. The third digit \(c\) can be any digit from 0 to 9 except for the digits chosen for \(a\) and \(b\) (8 options).
4. The fourth digit \(d\) can be any digit from 0 to 9 except for the digits chosen for \(a\), \(b\), and \(c\) (7 options).

The total number of 4-digit numbers is calculated as:
\[ 9 \times 9 \times 8 \times 7 = 4536 \]

### Step 2: Count 3-digit numbers
A 3-digit number can be represented as \(abc\).

1. The first digit \(a\) can be any digit from 1 to 9 (9 options).
2. The second digit \(b\) can be any digit from 0 to 9 except for the digit chosen for \(a\) (9 options).
3. The third digit \(c\) can be any digit from 0 to 9 except for the digits chosen for \(a\) and \(b\) (8 options).

The total number of 3-digit numbers is calculated as:
\[ 9 \times 9 \times 8 = 648 \]

### Step 3: Count 2-digit numbers
A 2-digit number can be represented as \(ab\).

1. The first digit \(a\) can be any digit from 1 to 9 (9 options).
2. The second digit \(b\) can be any digit from 0 to 9 except for the digit chosen for \(a\) (9 options).

The total number of 2-digit numbers is calculated as:
\[ 9 \times 9 = 81 \]

### Step 4: Count 1-digit numbers
A 1-digit number can be represented as \(a\).

1. The digit \(a\) can be any digit from 1 to 9 (9 options).

The total number of 1-digit numbers is:
\[ 9 \]

### Step 5: Add all the counts together
Now, we can sum all the numbers we have counted:
\[ 4536 \text{ (4-digit)} + 648 \text{ (3-digit)} + 81 \text{ (2-digit)} + 9 \text{ (1-digit)} = 5274 \]

Thus, the total number of natural numbers smaller than 10,000 with all different digits is **5274**.

### Final Answer
The total count is **5274**.