Question

Number of ways in which $7$ green bottles and $8$ blue bottles can be arranged in a row if exactly $1$ pair of green bottles is side by side is (Assume all bottles to be a like except for the colour).

• A. $84$
• B. $360$
• C. $504$
• D. None of the above

Answer 1

The correct option is C $504$

Consider the two green bottles as one entity. Let's call it $GG$.

We will also refer to the green bottle as: $G$, and to the blue bottle as: $B$.

So now we have $14$ units to arrange ( 1 green pair bottle + the remaining 5 green bottles + the 8 blue bottles):

$GG,G,G,G,G,G,B,B,B,B,B,B,B,B$

Now place the 8 blue bottles in a row, such that between each bottle has a gap before and after it:

_B_B_B_B_B_B_B_B_

Now we need to take the six green units (the 1 pair and 5 bottles) and place them in six of the nine gaps.

Therefore, the solution is: ${}^9{C}_6=84ways$.