Number of four digit numbers in which at least one digit occurs more than once, is :

Options

A. 4464
B. 4644
C. 4446
D. 6444

Answer

To solve the problem of finding the number of four-digit numbers in which at least one digit occurs more than once, we can use the principle of complementary counting. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the total number of four-digit numbers
A four-digit number cannot start with 0. Therefore, the first digit can be any digit from 1 to 9 (9 options). The remaining three digits can be any digit from 0 to 9 (10 options each).

Total four-digit numbers = (Choices for first digit) × (Choices for second digit) × (Choices for third digit) × (Choices for fourth digit)

\[ \text{Total four-digit numbers} = 9 \times 10 \times 10 \times 10 = 9000 \]

### Step 2: Calculate the number of four-digit numbers with all distinct digits
To find the number of four-digit numbers where no digit repeats, we again consider the restrictions:

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

Thus, the number of four-digit numbers with all distinct digits is:

\[ \text{Distinct four-digit numbers} = 9 \times 9 \times 8 \times 7 \]

Calculating this gives:

\[ 9 \times 9 = 81 \]
\[ 81 \times 8 = 648 \]
\[ 648 \times 7 = 4536 \]

### Step 3: Use complementary counting to find the desired count
Now, we can find the number of four-digit numbers in which at least one digit occurs more than once by subtracting the number of four-digit numbers with all distinct digits from the total number of four-digit numbers.

\[ \text{Numbers with at least one digit repeating} = \text{Total four-digit numbers} - \text{Distinct four-digit numbers} \]

Substituting the values we calculated:

\[ \text{Numbers with at least one digit repeating} = 9000 - 4536 = 4464 \]

### Final Answer
The number of four-digit numbers in which at least one digit occurs more than once is **4464**.

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