Problem Solving

How many whole numbers between 100 and 400, inclusive, contain the digit 2?

A. 100
B. 125
C. 138
D. 145
E. 150

Max@Math Revolution wrote:How many whole numbers between 100 and 400, inclusive, contain the digit 2?

A. 100
B. 125
C. 138
D. 145
E. 150

(integers with the digit 2) = (all integers between 100 and 399, inclusive) - (integers without the digit 2)

All integers between 100 and 399, inclusive:
For any set of consecutive integers, count = biggest - smallest + 1.
In this case:
count = 399-100+1 = 300

Integers without the digit 2:
Number of options for the hundreds digit = 2. (1 or 3)
Number of options for the tens digit = 9. (Any digit but 2)
Number of options for the units digit = 9. (Any digit but 2)
To combine these options, we multiply:
2*9*9 = 162

Integers with the digit 2:
300-162 = 138

The correct answer is C.

=>

There are 19 such numbers with hundreds digit 1: 102, 112, 120, 121, ..., 129, 132, 142, ..., 192.
There are 100 such numbers with hundreds digit 2: 200, 201, 202, 203, ..., 299.
There are 19 such numbers with hundreds digit 3: 302, 312, 320, 321, ..., 329, 332, 342, ..., 392.

Thus, there are a total of 19 + 100 + 19 = 138 such numbers.

Therefore, the answer is C.
Answer: C