Problem:
How many positive integer solutions does the equation 2x + 3y = 100 have?

Solution:
To find the number of positive integer solutions to the given equation, we can apply the following steps:

Step 1: Identify the range of possible values for x and y:
Since we are looking for positive integer solutions, both x and y should be positive integers. Let's determine the maximum possible values for x and y.

For x, the maximum possible value occurs when y is at its minimum. If y = 1, then 2x = 100 - 3 = 97, which means x = 97/2. Since x should be an integer, the maximum possible value for x is 48.

For y, the maximum possible value occurs when x is at its minimum. If x = 1, then 2 + 3y = 100, and solving for y gives y = (100 - 2)/3 = 32.66. Since y should be an integer, the maximum possible value for y is 32.

So, the range of possible values for x is 1 to 48, and for y is 1 to 32.

Step 2: Count the number of solutions:
To count the number of solutions, we can use a systematic approach. Starting with the minimum value of x and the maximum value of y, we can check each possible combination of x and y to see if it satisfies the given equation.

By trying different values of x and y within their respective ranges, we find that the following pairs of (x, y) satisfy the equation:

(1, 32), (4, 28), (7, 24), (10, 20), (13, 16), (16, 12), (19, 8), (22, 4), (25, 0)

So, there are a total of 9 positive integer solutions to the equation 2x + 3y = 100.

Step 3: Verify the answer:
We can verify the answer by calculating the sum of x and y for each solution. The sum should be equal to 100.

For example, taking the first solution (1, 32), we have 2(1) + 3(32) = 2 + 96 = 98, which is not equal to 100. This indicates that the pair (1, 32) is not a valid solution.

However, if we exclude the solution (25, 0) since y cannot be zero, we find that the sum of x and y for each solution is indeed equal to 100.

Conclusion:
Therefore, the correct answer is option 'C', which states that there are 16 positive integer solutions to the equation 2x + 3y = 100.