To solve this problem, we need to determine the number of possible groups of size 7, selected from a class of 10 freshmen and 8 seniors, such that the group contains at least 3 freshmen.

Let $F$ be the number of freshmen chosen, and $S$ be the number of seniors chosen. The conditions are:

1. The total number in the group is 7: $F + S = 7$.
2. The group must have at least 3 freshmen: $F \geq 3$.

We can explore the possibilities for $F$ and $S$ that satisfy both conditions and then calculate the combinations for each case.

Step 1: Determine Possible Combinations of Freshmen and Seniors

The possible values for $F$ (the number of freshmen) are 3, 4, 5, 6, and 7. For each choice of $F$, the number of seniors $S$ is determined by $S = 7 - F$.

Step 2: Calculate Combinations for Each Case

We’ll calculate the number of ways to choose $F$ freshmen from 10 and $S$ seniors from 8 for each valid combination of $F$ and $S$ using the combination formula: $\binom{n}{k} = \frac{n!}{k!(n - k)!}$

Case 1: $F = 3$ and $S = 4$

$\text{Ways} = \binom{10}{3} \times \binom{8}{4}$

Case 2: $F = 4$ and $S = 3$

$\text{Ways} = \binom{10}{4} \times \binom{8}{3}$

Case 3: $F = 5$ and $S = 2$

$\text{Ways} = \binom{10}{5} \times \binom{8}{2}$

Case 4: $F = 6$ and $S = 1$

$\text{Ways} = \binom{10}{6} \times \binom{8}{1}$

Case 5: $F = 7$ and $S = 0$

$\text{Ways} = \binom{10}{7} \times \binom{8}{0}$

Step 3: Perform Calculations

Let’s calculate each of these values:

1. Case 1: $\binom{10}{3} \times \binom{8}{4} = 120 \times 70 = 8400$
2. Case 2: $\binom{10}{4} \times \binom{8}{3} = 210 \times 56 = 11760$
3. Case 3: $\binom{10}{5} \times \binom{8}{2} = 252 \times 28 = 7056$
4. Case 4: $\binom{10}{6} \times \binom{8}{1} = 210 \times 8 = 1680$
5. Case 5: $\binom{10}{7} \times \binom{8}{0} = 120 \times 1 = 120$

Step 4: Sum All Cases

Now, we add up all the possibilities: $8400 + 11760 + 7056 + 1680 + 120 = 29116$

Conclusion

The total number of possible groups of 7 with at least 3 freshmen is: $\boxed{29116}$

Would you like more details on any part of this solution?

Here are 5 follow-up questions you might explore:

1. How would the solution change if the group had to include at least 4 freshmen?
2. What if the total number of students in the group was 6 instead of 7?
3. How many groups could be formed if there was no restriction on the minimum number of freshmen?
4. What would the answer be if the class included a different number of freshmen and seniors?
5. Can you derive a general formula for any group size and class composition?

Tip: When working with combination problems, carefully count cases to avoid overcounting or missing possible combinations.