Question

The number of positive integers that have at most seven digits and contain only the digits 0 and 9 is

• A. 112
• B. 127
• C. 136
• D. 142
• E. 150

Hint:

To find how many numbers can be formed with a certain number of digits using specific digits (like 0 and 9), use the power of choices for each digit.

Answer 1

The Correct Option is B

For each number, the digits can either be 0 or 9, and there are 7 digits at most. 
We calculate the total number of such numbers for 1, 2, ..., 7 digits. 
- For 1 digit: Only 1 possibility (9). 
- For 2 digits: 2 choices for the first digit (9 or 0, but first digit cannot be 0), and 2 choices for the second (9 or 0), so \( 2 \times 2 = 4 \). 
- For 3 digits: Similarly, \( 2 \times 2 \times 2 = 8 \), and so on. 
The total number of such numbers is \( 1 + 4 + 8 + 16 + 32 + 64 + 64 = 127 \). 
Thus, the correct answer is (B).