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:
- 4 pianists (exactly 2 to be selected)
- 5 violinists (exactly 3 or more to be selected)
- 3 guitarists (exactly 2 to be selected)
- 6 male singers
- 4 female singers

### Step 1: Choose the pianists
You need to choose exactly 2 out of 4 pianists.

The number of ways to do this is given by the combination formula \( \binom{n}{k} \):

\[
\text{Number of ways to choose pianists} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
\]

### Step 2: Choose the guitarists
You need to choose exactly 2 out of 3 guitarists.

The number of ways to do this is:

\[
\text{Number of ways to choose guitarists} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3
\]

### Step 3: Choose the violinists
You need to choose at least 3 out of 5 violinists. 

There are two possible scenarios:
1. Select exactly 3 violinists.
2. Select exactly 4 violinists.
3. Select all 5 violinists.

For each scenario, the number of ways is calculated as follows:

\[
\text{Number of ways to choose 3 violinists} = \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
\]

\[
\text{Number of ways to choose 4 violinists} = \binom{5}{4} = 5
\]

\[
\text{Number of ways to choose 5 violinists} = \binom{5}{5} = 1
\]

### Step 4: Choose the remaining members
After selecting the pianists, guitarists, and violinists, we calculate how many members remain to be selected to make a group of 11 members.

#### Case 1: 3 violinists are selected
- 2 pianists + 2 guitarists + 3 violinists = 7 members selected.
- Remaining members needed = 11 - 7 = 4 members from 10 singers (6 male + 4 female).

The number of ways to choose 4 members from 10 singers:

\[
\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]

#### Case 2: 4 violinists are selected
- 2 pianists + 2 guitarists + 4 violinists = 8 members selected.
- Remaining members needed = 11 - 8 = 3 members from 10 singers.

The number of ways to choose 3 members from 10 singers

Learn how to form groups of 11 members by selecting exactly 2 pianists, 2 guitarists, and at least 3 violinists. This problem involves combinatorial selection and uses the binomial coefficient formula extensively. Suitable for advanced high school students.

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