There are 6 boxes numbered 1, 2, 3,...6. Each box is to be..

[BTGmoderatorLU]

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?

A. 5
B. 21
C. 33
D. 60
E. 40

The OA is B.

Experts, I obtain the solution of the following way,

1G - 6ways
2G - 5ways
3G - 4 ways
4G - 3ways
5G - 2ways
6G - 1way

The total ways are 6 + 5 + 4 + 3 + 2 + 1 = 21 ways.

Experts, any suggestion? Thanks in advance.

[deloitte247]

If there is one box that contains a red or green ball, there will be 6 ways to do it. i,e 1,2,3,4,5,6.
so, if there are 2 boxes that contain red or green balls, there will be 5 ways to do it (1,2) (2,3)(3,4)(4,5) or (5,6).
for 3 boxes, there will be 4 ways, (1,2,3)(2,3,4)(3,4,5) or (4,5,6)
for 5 boxes, there will be 2 ways (1,2,3,4,5) or (2,3,4,5,6)
for 6 boxes, there is only one way to do it (1,2,3,4,5,6)

The total=1+2+3+4+5+6= 21 ways

Option B is correct

[Scott@TargetTestPrep]

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?

A. 5
B. 21
C. 33
D. 60
E. 40

If exactly 1 box has a green ball, there are 6C1 = 6 ways.

If exactly 2 boxes have a green ball, we can treat the 2 boxes as a single box since the 2 boxes are consecutively numbered, and hence there are 5C1 = 5 ways.

If exactly 3 boxes have a green ball, we can treat the 3 boxes as a single box since the 3 boxes are consecutively numbered, and hence there are 4C1 = 4 ways.

If exactly 4 boxes have a green ball, we can treat the 4 boxes as a single box since the 4 boxes are consecutively numbered, and hence there are 3C1 = 3 ways.

If exactly 5 boxes have a green ball, we can treat the 5 boxes as a single box since the 5 boxes are consecutively numbered, and hence there are 2C1 = 2 ways.

If all 6 boxes have a green ball, we can treat the 6 boxes as a single box since the 6 boxes are consecutively numbered, and hence there is 1C1 = 1 way.

Therefore, the total number of ways is 6 + 5 + 4 + 3 + 2 + 1 = 21.

Answer: B