There are 4 pianists, 5 violinists, 3 guitarists, 6 male singers and 4 female singers. Among
them, exactly 2 pianists, exactly 2 guitarists and minimum of 3 violinists are to be selected
for a group of 11 members. How many such groups can be formed by including the
members according to the above instructions?

To solve this problem, let's break it down step by step.

### Given:
- **Pianists**: 4
- **Violinists**: 5
- **Guitarists**: 3
- **Male Singers**: 6
- **Female Singers**: 4

### Requirements:
1. **Exactly 2 pianists** must be selected.
2. **Exactly 2 guitarists** must be selected.
3. **At least 3 violinists** must be selected.
4. A total of **11 members** must be selected.

### Step 1: Selecting Pianists
You need to select exactly 2 pianists from 4.

- The number of ways to do this is given by the combination formula:
\[
\text{Number of ways} = \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
\]

### Step 2: Selecting Guitarists
You need to select exactly 2 guitarists from 3.

- The number of ways to do this is given by:
\[
\text{Number of ways} = \binom{3}{2} = \frac{3 \times 2}{2 \times 1} = 3
\]

### Step 3: Selecting Violinists
You need to select at least 3 violinists from 5. There are two possible cases: selecting 3 violinists or selecting all 5 violinists.

- **Case 1: Selecting 3 violinists**:
\[
\text{Number of ways} = \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
\]

- **Case 2: Selecting 4 violinists**:
\[
\text{Number of ways} = \binom{5}{4} = \frac{5 \times 4}{2 \times 1} = 5
\]

- **Case 3: Selecting 5 violinists**:
\[
\text{Number of ways} = \binom{5}{5} = 1
\]

### Step 4: Selecting Singers
After selecting pianists, guitarists, and violinists, we need to fill the remaining spots with singers.

- **Case 1**: If 3 violinists are selected, then 4 more members (singers) are needed.
- **Case 2**: If 4 violinists are selected, then 3 more members (singers) are needed.
- **Case 3**: If 5 violinists are selected, then 2 more members (singers) are needed.

For each case, we calculate the number of ways to choose the remaining members from the 10 singers (6 male + 4 female).

- **Case 1: Selecting 3 violinists (4 singers needed)**:
\[
\text{Number of ways} = \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]

- **Case 2: Selecting 4 violinists (3 singers needed)**:
\[
\text{Number of ways} = \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]

- **Case 3: Selecting 5 violinists (2 singers needed)**:
\[
\text{Number of ways} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45
\]

### Step 5: Total Number of Groups
Now, we combine the cases:

1. **Case 1: 3 violinists + 4 singers**:
\[
6 \times 3 \times 10 \times 210 = 37800 \text{ groups}
\]

2. **Case 2: 4 violinists + 3 singers**:
\[
6 \times 3 \times 5 \times 120 = 10800 \text{ groups}
\]

3. **Case 3: 5 violinists + 2 singers**:
\[
6 \times 3 \times 1 \times 45 = 810 \text{ groups}
\]

### Final Answer:
Total number of possible groups:
\[
37800 + 10800 + 810 = 49410 \text{ groups}
\]

So, the total number of such groups that can be formed is **49,410**.

Do you want more details or have any questions?

---

**Related Questions:**
1. How do combinations differ from permutations?
2. What is the general formula for combinations?
3. How would the solution change if at least 2 female singers were required?
4. How can you calculate the probability of a specific group being selected?
5. What are some real-life applications of combinatorial selection problems?

**Tip:** In combinatorics, always make sure to identify whether the order of selection matters. This will determine whether you use combinations or permutations.

Explore how to solve complex combinatorial selection problems involving pianists, violinists, guitarists, male and female singers. This detailed solution includes selecting exact numbers of each group while meeting specific criteria. Suitable for advanced high school students and beyond.

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