In a group of 250 students, the percentage of girls was at least 44% and at most 60%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 80% of the girls opted for swimming while 70% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running are:

The correct option is 72 and 80, respectively.

Let us break down the solution step-by-step to understand how to arrive at this answer.

Step 1: Identify the variables and bounds for the number of students
Let the total number of students be $N=250$.
Let $G$ represent the number of girls and $B$ represent the number of boys. Thus:
$B+G=250$
The percentage of girls in the group is at least 44% and at most 60%. We can calculate the minimum and maximum number of girls as follows:
$G_{\text{min}}=44\%\text{of}250=0.44\times 250=110$
$G_{\text{max}}=60\%\text{of}250=0.60\times 250=150$
Therefore, the number of girls $G$ must satisfy the inequality:
$110\le G\le 150$

Step 2: Determine the number of boys who opted for both activities
We are given that each student opted for either swimming, running, or both. This means there are no students who did not opt for at least one of the two activities.
For boys:
- Percentage opting for swimming = 50%
- Percentage opting for running = 70%
Let $B_{\text{both}}$ be the number of boys who opted for both swimming and running. Using the principle of inclusion-exclusion for sets:
$B=0.5B+0.7B-B_{\text{both}}$
Rearranging the equation to solve for $B_{\text{both}}$:
$B_{\text{both}}=1.2B-B=0.2B$
Since the number of boys opting for both must be an integer, $0.2B$ must be an integer. This implies that $B$ must be a multiple of 5. Since $B=250-G$, the number of girls $G$ must also be a multiple of 5.

Step 3: Determine the number of girls who opted for both activities
For girls:
- Percentage opting for swimming = 80%
- Percentage opting for running = 60%
Let $G_{\text{both}}$ be the number of girls who opted for both swimming and running. Using the set inclusion-exclusion principle:
$G=0.8G+0.6G-G_{\text{both}}$
Rearranging the equation to solve for $G_{\text{both}}$:
$G_{\text{both}}=1.4G-G=0.4G$
Since $G$ is a multiple of 5, $G_{\text{both}}=0.4G$ will always evaluate to an integer value.

Step 4: Calculate the total number of students opting for both activities
The total number of students who opted for both swimming and running is:
$T_{\text{both}}=B_{\text{both}}+G_{\text{both}}$
Substituting the expressions we derived in terms of $B$ and $G$:
$T_{\text{both}}=0.2B+0.4G$
Substitute $B=250-G$ into the equation:
$T_{\text{both}}=0.2(250-G)+0.4G$
$T_{\text{both}}=50-0.2G+0.4G$
$T_{\text{both}}=50+0.2G$

Step 5: Find the minimum and maximum possible values
Since $T_{\text{both}}$ is a strictly increasing function of the number of girls ($G$):
- The minimum value of $T_{\text{both}}$ occurs when $G$ is at its minimum ($G=110$):
$T_{\text{both, min}}=50+0.2\times 110=50+22=72$
- The maximum value of $T_{\text{both}}$ occurs when $G$ is at its maximum ($G=150$):
$T_{\text{both, max}}=50+0.2\times 150=50+30=80$
Thus, the minimum and maximum possible number of students who opted for both swimming and running are 72 and 80, respectively.