To find the total number of numbers that are divisible by 2 or 3 between 100 and 200 (both inclusive), we can use the concept of the inclusion-exclusion principle.
First, let's find the number of numbers divisible by 2 between 100 and 200.
The first number divisible by 2 in this range is 100, and the last number divisible by 2 is 200. We can find the count of numbers divisible by 2 by finding the number of odd numbers in this range.
The number of odd numbers between 100 and 200 (both inclusive) can be calculated as:
[ \frac{{\text{{last number}} - \text{{first number}}}}{2} + 1 = \frac{{200 - 100}}{2} + 1 = 51 ]
Next, let's find the number of numbers divisible by 3 between 100 and 200.
The first number divisible by 3 in this range is 102, and the last number divisible by 3 is 198. We can find the count of numbers divisible by 3 by dividing the range by 3 and adding 1.
The number of numbers divisible by 3 between 100 and 200 (both inclusive) can be calculated as:
[ \frac{{\text{{last number}} - \text{{first number}}}}{3} + 1 = \frac{{198 - 102}}{3} + 1 = 33 ]
However, we need to consider that some numbers are counted twice since they are divisible by both 2 and 3. To find the count of these numbers, we can divide the range by the least common multiple of 2 and 3, which is 6, and add 1.
The number of numbers divisible by both 2 and 3 between 100 and 200 (both inclusive) can be calculated as:
[ \frac{{\text{{last number}} - \text{{first number}}}}{6} + 1 = \frac{{198 - 102}}{6} + 1 = 17 ]
Now, let's apply the inclusion-exclusion principle to find the total count of numbers divisible by 2 or 3. We can calculate this by adding the count of numbers divisible by 2 to the count of numbers divisible by 3 and subtracting the count of numbers divisible by both 2 and 3.
Total count of numbers divisible by 2 or 3 between 100 and 200 (both inclusive) can be calculated as:
[ 51 + 33 - 17 = 67 ]
Therefore, the correct answer is option B) 67.