To solve the problem of selecting 7 subjects from 3 different groups A, B, and C, where group A has 4 subjects, group B has 5 subjects, and group C has 6 subjects, and the condition is to select at least 2 subjects from each group, we can follow these steps:
### Step 1: Define the Selection Criteria
We need to select a total of 7 subjects while ensuring that at least 2 subjects are chosen from each group. This means we can break down the selection into different cases based on how many subjects we choose from each group.
### Step 2: Determine the Cases
Since we need at least 2 subjects from each group, we can denote the number of subjects selected from each group as follows:
- Let \( x_1 \) be the number of subjects chosen from group A
- Let \( x_2 \) be the number of subjects chosen from group B
- Let \( x_3 \) be the number of subjects chosen from group C
We know:
- \( x_1 + x_2 + x_3 = 7 \)
- \( x_1 \geq 2 \), \( x_2 \geq 2 \), \( x_3 \geq 2 \)
### Step 3: Adjust the Variables
To simplify, we can set:
- \( y_1 = x_1 - 2 \)
- \( y_2 = x_2 - 2 \)
- \( y_3 = x_3 - 2 \)
Now, the equation becomes:
- \( y_1 + y_2 + y_3 = 1 \)
- \( y_1 \geq 0 \), \( y_2 \geq 0 \), \( y_3 \geq 0 \)
### Step 4: Find the Combinations
The possible distributions of subjects can be:
1. \( (2, 2, 3) \)
2. \( (2, 3, 2) \)
3. \( (3, 2, 2) \)
### Step 5: Calculate Each Case
Now we calculate the number of ways to select subjects for each case:
#### Case 1: \( (2, 2, 3) \)
- Choose 2 from A (4 subjects): \( \binom{4}{2} = 6 \)
- Choose 2 from B (5 subjects): \( \binom{5}{2} = 10 \)
- Choose 3 from C (6 subjects): \( \binom{6}{3} = 20 \)
Total for this case:
\[ 6 \times 10 \times 20 = 1200 \]
#### Case 2: \( (2, 3, 2) \)
- Choose 2 from A (4 subjects): \( \binom{4}{2} = 6 \)
- Choose 3 from B (5 subjects): \( \binom{5}{3} = 10 \)
- Choose 2 from C (6 subjects): \( \binom{6}{2} = 15 \)
Total for this case:
\[ 6 \times 10 \times 15 = 900 \]
#### Case 3: \( (3, 2, 2) \)
- Choose 3 from A (4 subjects): \( \binom{4}{3} = 4 \)
- Choose 2 from B (5 subjects): \( \binom{5}{2} = 10 \)
- Choose 2 from C (6 subjects): \( \binom{6}{2} = 15 \)
Total for this case:
\[ 4 \times 10 \times 15 = 600 \]
### Step 6: Sum All Cases
Now, we sum the totals from all cases:
\[ 1200 + 900 + 600 = 2700 \]
### Final Answer
Therefore, the total number of different ways a candidate can make his selection is **2700**.
