there are 5 different boxes and 7 different balls. All the 7 balls are to be distributed in the 5 boxes placed in a row so that any box can receive any number of balls .
In the previous question (no. 70) in how many ways can all these balls be distributed into these boxes so that no box remains empty and no two consecutive boxes have the same number of balls?

Options

A. 1
B. 4
C. 12
D. none of these

Answer

To solve the problem of distributing 7 different balls into 5 different boxes such that no box remains empty and no two consecutive boxes have the same number of balls, we can follow these steps:

### Step 1: Understand the Problem
We have 5 boxes and 7 balls. We need to distribute the balls in such a way that:
1. No box is empty.
2. No two consecutive boxes have the same number of balls.

### Step 2: Assign Minimum Balls to Each Box
Since no box can be empty and we have 5 boxes, we must assign at least 1 ball to each box. This uses up 5 balls, leaving us with 2 balls to distribute freely among the boxes.

### Step 3: Distributing the Remaining Balls
Now we have 2 remaining balls to distribute among the 5 boxes. However, we must ensure that no two consecutive boxes have the same number of balls.

### Step 4: Determine Possible Distributions
Let’s denote the number of balls in each box as \( b_1, b_2, b_3, b_4, b_5 \). We know:
- \( b_1 + b_2 + b_3 + b_4 + b_5 = 7 \)
- \( b_i \geq 1 \) for all \( i \)
- \( b_i \neq b_{i+1} \) for all \( i \)

To satisfy the condition of no two consecutive boxes having the same number of balls, we can start with the minimum distribution of balls.

### Step 5: Possible Arrangements
1. Start with the arrangement:
- Box 1: 1 ball
- Box 2: 2 balls
- Box 3: 1 ball
- Box 4: 2 balls
- Box 5: 1 ball

This arrangement gives us a total of 1 + 2 + 1 + 2 + 1 = 7 balls, and no two consecutive boxes have the same number of balls.

2. Check other arrangements:
- If we try to assign 2 balls to Box 1, we cannot assign 2 balls to Box 2 (as that would violate the consecutive condition). The only option would be to assign 1 ball to Box 2, which leads to a violation of the total balls constraint.

### Step 6: Conclusion
After checking possible distributions, we find that the only valid arrangement is:
- Box 1: 1 ball
- Box 2: 2 balls
- Box 3: 1 ball
- Box 4: 2 balls
- Box 5: 1 ball

Thus, there is only **1 way** to distribute the balls under the given conditions.

### Final Answer
The number of ways to distribute the balls is **1**.

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