Question

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. $60$

Answer 1

Answer: (B)

$21$

(B) The number of ways in which 1 green ball can be put $=6$ . The number of ways in which two green balls can be put such that the boxes are consecutive
$=5$ $(i.e.,(1,2),(2,3),(3,4),(4,5),(5,6\left)\right)$
Similarly, the number of ways in which three green balls can be put
$=4(i.e.(1,2,3),(2,3,4),(3,4,5),(4,5,6\left)\right)$
$\cdots \cdots \cdots \cdots \cdots$ and so on.
$\therefore$ Total number of ways of doing this
$=6+5+4+3+2+1=21$