In how many ways 7 persons can be selected from among 5 Indian, 4 British & 2 Chinese, If atleast two are to be selected from each country?

Answer

To solve the problem of selecting 7 persons from 5 Indians, 4 British, and 2 Chinese with the condition that at least 2 persons must be selected from each nationality, we can break it down into manageable steps.

### Step-by-Step Solution:

1. **Understanding the Selection Requirement**:
We need to select a total of 7 persons such that at least 2 are from each nationality. Given that we have:
- 5 Indians
- 4 British
- 2 Chinese

2. **Fixing the Chinese Selection**:
Since we must select at least 2 Chinese, we will start by selecting 2 Chinese. This leaves us with:
- Total persons to select = 7
- Persons already selected = 2 (Chinese)
- Remaining persons to select = 7 - 2 = 5

3. **Distributing the Remaining Selections**:
Now we need to select 5 persons from the remaining nationalities (Indians and British) with the condition that at least 2 must be selected from each of these two nationalities.

4. **Case Analysis**:
We can have two cases for the selection of Indians and British:
- **Case 1**: Select 2 Indians and 3 British.
- **Case 2**: Select 3 Indians and 2 British.

5. **Calculating Case 1 (2 Indians and 3 British)**:
- Number of ways to select 2 Indians from 5:
\[ \binom{5}{2} = 10 \]
- Number of ways to select 3 British from 4:
\[ \binom{4}{3} = 4 \]
- Total ways for Case 1:
\[ 10 \times 4 = 40 \]

6. **Calculating Case 2 (3 Indians and 2 British)**:
- Number of ways to select 3 Indians from 5:
\[ \binom{5}{3} = 10 \]
- Number of ways to select 2 British from 4:
\[ \binom{4}{2} = 6 \]
- Total ways for Case 2:
\[ 10 \times 6 = 60 \]

7. **Total Ways**:
Now, we add the number of ways from both cases:
\[ \text{Total ways} = 40 + 60 = 100 \]

### Final Answer:
Thus, the total number of ways to select 7 persons such that at least 2 are from each nationality is **100**.

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