There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is

Options

A. 5
B. 21
C. 33
D. 60

Answer

To solve the problem, we need to find the total number of ways to fill 6 boxes with either a red or a green ball, ensuring that at least one box contains a green ball and that all boxes containing green balls are consecutively numbered.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 6 boxes, and we can fill each box with either a red ball (R) or a green ball (G). The requirement is that at least one box must contain a green ball, and all green balls must be in consecutive boxes.

2. **Identifying the Groups of Boxes**: Since the green balls must be consecutive, we can think of them as forming a single block or group. For example, if boxes 2, 3, and 4 contain green balls, we can denote this as GGG (for boxes 2, 3, and 4) and the rest as R (for red balls).

3. **Calculating the Number of Ways to Place the Green Block**:
- The green block can start at box 1 and can extend to box 6.
- The possible lengths of the green block can be 1 to 6 boxes long.

4. **Counting the Positions**:
- If the green block has a length of 1, it can be placed in any of the 6 boxes (positions 1 to 6).
- If the green block has a length of 2, it can start in boxes 1 to 5 (positions 1-2, 2-3, 3-4, 4-5, 5-6).
- If the green block has a length of 3, it can start in boxes 1 to 4 (positions 1-3, 2-4, 3-5, 4-6).
- If the green block has a length of 4, it can start in boxes 1 to 3 (positions 1-4, 2-5, 3-6).
- If the green block has a length of 5, it can start in boxes 1 to 2 (positions 1-5, 2-6).
- If the green block has a length of 6, it can only be in position 1 (positions 1-6).

5. **Summing the Possible Positions**:
- Length 1: 6 ways
- Length 2: 5 ways
- Length 3: 4 ways
- Length 4: 3 ways
- Length 5: 2 ways
- Length 6: 1 way

Total = 6 + 5 + 4 + 3 + 2 + 1 = 21 ways.

6. **Final Result**: Therefore, the total number of ways to fill the boxes such that at least one box contains a green ball and all green balls are consecutively numbered is **21**.