To solve this problem, we need to determine the number of elements in each region $R_1$ through $R_5$ using the given information.

Given Information:

• $n(A) = 30$
• $n(B) = 33$
• $n(C) = 40$
• $n(A \cap B) = 15$
• $n(U) = 78$

Regions:

• $R_1$ represents the part of $A$ only.
• $R_2$ represents the intersection of $A$ and $B$ but not $C$.
• $R_3$ represents the intersection of $B$ and $C$ but not $A$.
• $R_4$ represents the intersection of $C$ and $A$ but not $B$.
• $R_5$ represents the part of $C$ only.
• The central region where all three circles intersect is already given as 3.

Step 1: Determine $R_2$

Given $n(A \cap B) = 15$ and the central region (intersection of $A$, $B$, and $C$) is 3, we can determine $R_2$:

$R_2 = n(A \cap B) - \text{central region} = 15 - 3 = 12$

Step 2: Determine $R_3$ and $R_4$

Since no direct intersections of $A \cap C$ or $B \cap C$ are provided, let’s assume $R_3$ and $R_4$ are labeled as in the diagram:

• $R_3$ is 5.
• $R_4$ is 7.

Step 3: Use $n(A)$, $n(B)$, $n(C)$ to find $R_1$ and $R_5$

For $R_1$ (part of $A$ only):

$R_1 = n(A) - (R_2 + \text{central region} + R_4) = 30 - (12 + 3 + 7) = 30 - 22 = 8$

For $R_5$ (part of $C$ only):

$R_5 = n(C) - (R_4 + R_3 + \text{central region}) = 40 - (7 + 5 + 3) = 40 - 15 = 25$

Step 4: Verification

Let’s verify by summing all regions and comparing with $n(U)$:

$n(U) = R_1 + R_2 + R_3 + R_4 + R_5 + \text{central region} = 8 + 12 + 5 + 7 + 25 + 3 = 60$

However, $n(U)$ is given as 78, so there’s an additional region outside $A$, $B$, and $C$ which contains:

$\text{Remaining region outside} = 78 - 60 = 18$

This implies there are 18 elements in the region outside all three circles.

Final Answer:

• $n(R_1) = 8$
• $n(R_2) = 12$
• $n(R_3) = 5$
• $n(R_4) = 7$
• $n(R_5) = 25$

Would you like more details on how these calculations were made, or do you have any further questions?

Related Questions:

1. How do you calculate the number of elements in each region for different sets if $n(A \cup B \cup C)$ was provided instead of $n(U)$?
2. What happens if an additional set $D$ is introduced to the Venn diagram?
3. How would the calculation change if the regions had overlapping elements not just in pairs but in triplets?
4. Can you generalize this approach to more than three sets? How would the calculations look?
5. What are the implications if $n(U)$ was significantly larger or smaller than the sum of individual sets?

Tip:

Always double-check each set's calculations by verifying the total count with $n(U)$ to ensure accuracy.