To calculate the total number of phone calls made during the month when the fifth member was added, we need to first understand how many members are in the club at that time and how the calls are made.
- Each member calls every other member in the club.
- The club starts with 2 members.
- They add 1 new member each month.
Let's list the number of members in the club for each month:
- Month 1: 2 members (initial)
- Month 2: 3 members (1 new member added)
- Month 3: 4 members (1 new member added)
- Month 4: 5 members (1 new member added)
- Month 5: 6 members (1 new member added)
We are interested in the fifth month when the club has 6 members.
Next, we compute the total number of calls made in this month. Each member calls every other member, which means we need to count the number of unique pairs of members that can make calls, since each pair results in one call from one member to the other.
The formula to find the number of unique pairs of \( n \) members is given by the binomial coefficient:
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
For \( n = 6 \) (as there are 6 members in the club during the fifth month):
\[ \binom{6}{2} = \frac{6 \times 5}{2} = \frac{30}{2} = 15 \]
So, there are 15 unique pairs, which means there will be 15 calls made in total during the month when the fifth member was added.
Thus, the total number of phone calls made during that month is \(\boxed{30}\). We should double this, since each member will call every other member once, resulting in one call each for both members in each unique pair, leading to:
\[ \text{Total calls} = 6 \times 5 = 30 \]
Therefore, the correct answer for the total number of phone calls made during the month in which the fifth member was added is indeed \(\boxed{30}\).