The problem involves dividing 11 members into 4 groups, with at least one member in each group. The total number of selected members will be 7. We need to find the number of ways to form this selection while ensuring all groups receive at least one member.

Here's how we can break down the problem:

Step 1: Assign at least one member to each group.

To satisfy the condition that each group must have at least one member, we first assign one member to each of the 4 groups. This leaves us with $11 - 4 = 7$ members to distribute freely.

Step 2: Distribute the remaining 7 members among the 4 groups.

Now we need to distribute the remaining 7 members among the 4 groups. This is a classical "stars and bars" problem, where we need to solve the equation: $x_1 + x_2 + x_3 + x_4 = 7$ where $x_i \geq 0$ represents the number of members assigned to the $i$-th group. This is equivalent to finding the number of non-negative integer solutions to this equation, which is given by the stars and bars formula: $\text{Number of solutions} = \binom{7 + 4 - 1}{4 - 1} = \binom{10}{3}$

Step 3: Calculate the combinations.

We now calculate $\binom{10}{3}$, which is the number of ways to distribute 7 remaining members among 4 groups: $\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$

Conclusion:

There are 120 different ways to divide 11 members into 4 groups, such that at least one member is in each group, and the total selected members are 7.

Would you like further details on any part of the solution or have other questions? Here are 5 questions for further exploration:

1. What if we need to select all 11 members instead of 7?
2. How does the stars and bars method apply to different group distributions?
3. How would the result change if we wanted at least 2 members in each group?
4. Can we calculate the probability of a specific member being in a particular group?
5. How does this apply to real-world scenarios, like team assignments?

Tip: Always verify conditions carefully when distributing objects into groups, especially when there are restrictions like "at least one in each group."