There are 150 students taking Discrete Mathematics II, Calculus II, and Physics I courses. Of these 51 are taking Discrete Mathematics II, 111 are taking Calculus II, and 63 are taking Physics I. There are 41 taking Discrete Mathematics II and Calculus II, 32 students taking Calculus II and Physics I, and 11 students taking Physics I and Discrete Mathematics II. How many of these students are taking exactly two of these three courses?

just make your usual Venn diagram and start labeling the intersections.

To find the number of students who are taking exactly two of the three courses, we need to subtract the number of students who are taking only one course and the number of students who are taking all three courses from the total number of students.

Let's break down the information given:

- Number of students taking Discrete Mathematics II (D): 51
- Number of students taking Calculus II (C): 111
- Number of students taking Physics I (P): 63
- Number of students taking Discrete Mathematics II and Calculus II (D ∩ C): 41
- Number of students taking Calculus II and Physics I (C ∩ P): 32
- Number of students taking Physics I and Discrete Mathematics II (P ∩ D): 11
- Total number of students (T): 150

Now let's calculate the number of students taking exactly two of the three courses:

Step 1: Calculate the number of students taking only one course.
- D - (D ∩ C) - (P ∩ D) = 51 - 41 - 11 = 51 - 52 = -1 (This means there is an overlap in the data)

Step 2: Calculate the number of students taking all three courses.
- We need to find the intersection of all three sets: D ∩ C ∩ P = ?

To find the intersection of all three sets, we can subtract the number of students not taking any of the courses from the total number of students (T):

- T - (D ∪ C ∪ P) = 150 - (51 + 111 + 63 - (D ∩ C) - (P ∩ D) - (C ∩ P)) = 150 - (51 + 111 + 63 - 41 - 11 - 32) = 150 - (277 - 84) = 150 - 193 = -43 (This means there is an overlap in the data)

Based on the given information, it seems there may be an error or overlap in the data. The calculated numbers for students taking only one course and students taking all three courses are negative, which is not feasible.

Please double-check the information or provide any additional details if available to resolve the inconsistency and provide an accurate answer.