If number of ways in which 7 different balls can be distributed into 4 different boxes, so that no box remains empty is `100lamda`, the value of `lamda` is

Options

A. 18
B. 108
C. 1008
D. 10008

Answer

To solve the problem of distributing 7 different balls into 4 different boxes such that no box remains empty, we can use the principle of inclusion-exclusion or Stirling numbers of the second kind. However, in this case, we can also break it down into cases based on the distribution of balls.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to distribute 7 different balls into 4 different boxes with the condition that no box is empty. This means each box must contain at least one ball.

2. **Using the Stars and Bars Method**:
To ensure that no box is empty, we can initially place one ball in each box. This uses up 4 balls, leaving us with \(7 - 4 = 3\) balls to distribute freely among the 4 boxes.

3. **Distributing Remaining Balls**:
We now need to distribute these 3 remaining balls into the 4 boxes. This can be done using the stars and bars method, where we need to find the number of non-negative integer solutions to the equation:
\[ x_1 + x_2 + x_3 + x_4 = 3 \]
where \(x_i\) represents the number of additional balls in box \(i\).

4. **Applying the Stars and Bars Theorem**:
The number of solutions to the equation \(x_1 + x_2 + x_3 + x_4 = 3\) is given by the formula:
\[ \binom{n+k-1}{k-1} \]
where \(n\) is the number of balls to distribute (3), and \(k\) is the number of boxes (4). Thus, we have:
\[ \binom{3+4-1}{4-1} = \binom{6}{3} = 20 \]

5. **Arranging the Balls**:
Since the balls are different, we must also consider the arrangements of the balls. The total number of arrangements of the 7 different balls is \(7!\).

6. **Total Ways to Distribute**:
Therefore, the total number of ways to distribute the balls is given by:
\[ 20 \times 7! = 20 \times 5040 = 100800 \]

7. **Finding the Value of Lambda**:
We are given that the total number of ways is \(100 \lambda\). Thus, we set up the equation:
\[ 100 \lambda = 100800 \]
Solving for \(\lambda\):
\[ \lambda = \frac{100800}{100} = 1008 \]

### Final Answer:
The value of \(\lambda\) is \(1008\).