Question

There are 6 boxes numbered 1, 2, 3, 4, 5, 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:

• A. 21
• B. 33
• C. 11
• D. 60
• E. 52

Hint:

For consecutive selections, the number of ways = total positions - length + 1.

Answer 1

The Correct Option is A

Step 1: Boxes with green balls must be consecutive. Let the number of green boxes be $k$, where $1 \le k \le 6$.
Step 2: For a given $k$, the number of ways to choose consecutive boxes = number of starting positions = $6 - k + 1 = 7 - k$.
Step 3: Total ways = $\sum_{k=1}^{6} (7 - k) = 6 + 5 + 4 + 3 + 2 + 1 = 21$.
Step 4: Final Answer: 21.