4 different balls of green colour and 4 different balls of red colour are to be distributed equally among 4 people have balls of a different colour is `lambda`, then the value of `7lambda` is equal to

Answer

To solve the problem, we need to find the value of \( 7\lambda \) where \( \lambda \) represents the number of ways to distribute 4 different green balls and 4 different red balls equally among 4 people.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 4 different green balls and 4 different red balls, making a total of 8 balls. We need to distribute these 8 balls equally among 4 people.

2. **Distribution of Balls**:
Each person will receive \( \frac{8}{4} = 2 \) balls. Since the balls are of different colors, we need to consider the arrangements of these balls.

3. **Calculating the Total Arrangements**:
The total number of ways to distribute the 8 balls can be calculated using the multinomial coefficient:
\[ \text{Total ways} = \frac{8!}{2! \times 2! \times 2! \times 2!} \]
Here, \( 8! \) accounts for the arrangements of all balls, and each \( 2! \) in the denominator accounts for the indistinguishable arrangements of the balls given to each person.

4. **Calculating \( \lambda \)**:
Since each person receives 2 balls, we can also think of this as distributing the green and red balls separately. The number of ways to distribute the green balls is:
\[ 4! \text{ (for green balls)} \]
and the number of ways to distribute the red balls is:
\[ 4! \text{ (for red balls)} \]
Thus, the total number of ways to distribute all balls is:
\[ \lambda = \frac{4! \times 4!}{2^4} \]

5. **Calculating \( \lambda \)**:
Now substituting the values:
\[ \lambda = \frac{24 \times 24}{16} = \frac{576}{16} = 36 \]

6. **Finding \( 7\lambda \)**:
Now we need to find \( 7\lambda \):
\[ 7\lambda = 7 \times 36 = 252 \]

### Final Answer:
The value of \( 7\lambda \) is \( 252 \).