How many numbers are there between 100 and 1000 such that at least one of their digits is 7 ?

Options

A. 900
B. 648
C. 252
D. 729

Answer

To solve the problem of how many numbers between 100 and 1000 have at least one digit as 7, we can use the complementary counting method. This means we will first calculate the total number of three-digit numbers in that range and then subtract the numbers that do not contain the digit 7 at all.

### Step 1: Calculate the total number of three-digit numbers between 100 and 1000.

The three-digit numbers range from 100 to 999. Therefore, the total number of three-digit numbers is:

\[ 999 - 100 + 1 = 900 \]

### Step 2: Calculate the number of three-digit numbers that do not contain the digit 7.

1. **Hundreds place**: The hundreds digit can be any digit from 1 to 9, except for 7. Therefore, the possible choices for the hundreds place are: 1, 2, 3, 4, 5, 6, 8, 9. This gives us 8 options.

2. **Tens place**: The tens digit can be any digit from 0 to 9, except for 7. Therefore, the possible choices for the tens place are: 0, 1, 2, 3, 4, 5, 6, 8, 9. This gives us 9 options.

3. **Units place**: The units digit can also be any digit from 0 to 9, except for 7. Therefore, the possible choices for the units place are: 0, 1, 2, 3, 4, 5, 6, 8, 9. This gives us 9 options.

Now, we multiply the number of choices for each digit:

\[ \text{Total without 7} = 8 \times 9 \times 9 = 648 \]

### Step 3: Calculate the number of three-digit numbers that contain at least one digit as 7.

To find the numbers that contain at least one digit as 7, we subtract the count of numbers without any 7s from the total count of three-digit numbers:

\[ \text{Total with at least one 7} = \text{Total three-digit numbers} - \text{Total without 7} \]

\[ \text{Total with at least one 7} = 900 - 648 = 252 \]

### Final Answer

Thus, the total number of numbers between 100 and 1000 such that at least one of their digits is 7 is **252**.

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