We are tasked with finding the number of elements in each of the five regions $R_1, R_2, R_3, R_4, R_5$ using the provided set information.

Here are the known values:

• $n(A) = 29$ (number of elements in set $A$),
• $n(B) = 33$ (number of elements in set $B$),
• $n(C) = 39$ (number of elements in set $C$),
• $n(A \cap B) = 15$ (number of elements common to both $A$ and $B$),
• $n(U) = 79$ (the total number of elements in the universal set $U$).

To clarify, we are dealing with three overlapping sets $A$, $B$, and $C$, where:

• $R_1$: elements in $A$ only,
• $R_2$: elements in $A \cap B$ (common to both $A$ and $B$ but not in $C$),
• $R_3$: elements in $B$ only,
• $R_4$: elements in $C$,
• $R_5$: elements outside of $A$, $B$, and $C$.

We will now proceed to find these values step by step.

1. $n(A \cup B)$: First, we use the formula for the union of two sets: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ Substitute the given values: $n(A \cup B) = 29 + 33 - 15 = 47$ So, there are 47 elements in either $A$, $B$, or both.

2. $n(C)$: There are 39 elements in set $C$, but we are also interested in the overlap with $A \cup B$. However, no information is provided for the overlap of $C$ with $A$ or $B$ directly. Since the total number of elements in the universal set $U$ is given as 79, and we know the number of elements in $A \cup B$, we can find $R_5$.

3. $n(R_5)$: This is the number of elements outside $A$, $B$, and $C$, i.e., elements in the universal set $U$ but not in any of the sets $A$, $B$, or $C$: $n(R_5) = n(U) - n(A \cup B \cup C)$ At this point, we would need the additional information of how $C$ overlaps with $A$ and $B$ to complete the solution fully.

Would you like me to proceed further with assumptions or clarify any points?