Use the given information to find the number of elements in each of the regions labeled R1 through R5.
n(A) = 30, n(B) = 33, n(C) = 36, n(A ∩ B) = 16, n(U) = 75

What is R5?

Let's break down the problem systematically.

We are given:

- \( n(A) = 30 \) (The number of elements in set \( A \))
- \( n(B) = 33 \) (The number of elements in set \( B \))
- \( n(C) = 36 \) (The number of elements in set \( C \))
- \( n(A \cap B) = 16 \) (The number of elements in the intersection of sets \( A \) and \( B \))
- \( n(U) = 75 \) (The total number of elements in the universal set \( U \))

### Steps to find \( R_5 \):

The regions labeled \( R_1, R_2, R_3, R_4, R_5 \) correspond to different parts of the Venn diagram formed by sets \( A \), \( B \), and \( C \). We can assume the following:

- \( R_1 \) represents elements in only \( A \) (not in \( B \) or \( C \)).
- \( R_2 \) represents elements in only \( B \) (not in \( A \) or \( C \)).
- \( R_3 \) represents elements in only \( C \) (not in \( A \) or \( B \)).
- \( R_4 \) represents elements in \( A \cap B \cap C \) (intersection of all three sets).
- \( R_5 \) represents elements in neither \( A \), \( B \), nor \( C \) (outside of all three sets).

Since \( R_5 \) represents elements outside of all three sets, it can be calculated using:

\[
R_5 = n(U) - (n(A) + n(B) + n(C) - n(A \cap B))
\]

This formula accounts for the overlap between \( A \) and \( B \), preventing double-counting the elements.

Let me compute this for you.It appears that using the formula as stated results in a negative value, which is not possible. This indicates a problem with the approach or the interpretation of the set intersections.

Let's reconsider the structure of the problem.

We know:
- \( n(A \cap B) = 16 \), the intersection of sets \( A \) and \( B \), but we do not have specific information about intersections with \( C \), nor about other specific intersections (such as \( A \cap C \) or \( B \cap C \)).
- We are looking for the number of elements outside sets \( A \), \( B \), and \( C \), which are counted by \( R_5 \).

However, the error suggests that there's missing or incorrect information about intersections, specifically about overlaps with \( C \). With more details on those intersections, we could correct the calculation. Would you happen to have more information on intersections involving \( C \)?

Use the given information to find the number of elements in each of the regions labeled R1 through R5. n(A) = 30, n(B) = 33, n(C) = 36, n(A ∩ B) = 16, n(U) = 75. What is R5?

Learn how to calculate the number of elements in the region R5 of a Venn diagram with three sets (A, B, and C). This step-by-step solution explains key concepts in set theory, including intersections, universal sets, and the principle of inclusion-exclusion. Suitable for students in Grades 9-12.

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