# Difficult Permutation-Combination Solved QuestionAptitude Discussion

 Q. 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. 27 ✖ E. 60

Solution:
Option(B) is correct

If only one of the boxes has a green ball, it can be any of the 6 boxes. So, this can be achieved in $6$ ways.

If two of the boxes have green balls and then there are $5$ arrangement possible. i.e., the two boxes can one of $1-2$ or $2-3$ or $3-4$ or $4-5$ or $5-6$.

If $3$ of the boxes have green balls, there will be $4$ options in which the $3$ boxes are in consecutive positions. i.e., $1-2-3$ or $2-3-4$ or $3-4-5$ or $4-5-6$

If $4$ boxes have green balls, there will be $3$ options. i.e., $1-2-3-4$ or $2-3-4-5$ or $3-4-5-6$

If $5$ boxes have green balls, then there will be $2$ options. i.e., $1-2-3-4-5$ or $2-3-4-5-6$

If all $6$ boxes have green balls, then there will be just $1$ options.

Total number of options,

$= 6 + 5 + 4 + 3 + 2 + 1$

$= \textbf{21}$

## (2) Comment(s)

Manohar Tangi
()

the boxes containing green balls are consecutively numbered.

what exactly it means?

and why can't,6-1 or 5-6-1 and so on..

Manohar Tangi
()

the boxes containing green balls are consecutively numbered.

what exactly it means?