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 ______
Note-Assume all bottles to be alike except for the color.
Attempt
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$8$ Blue bottles firstly arranged in $1$ ways only since all are identical. Now selected $2$ green bottles in $1$ way only again as it is identical. Now the $6$ elements ($1$ pair bottles and $5$ bottles green in color) can be placed into $^9C_6$ which is equal to $84$ ways. But the answer in answer key is $504$ ways.
Where is the fault in my process? I tried to think about the flaw but couldn't encounter on it.
Best Answer
Doing it with $\binom96$ you overlook that the spot of the $2$ green bottles matters.
E.g. bggbgbgbgbgbgbb and bgbggbgbgbgbgbb are essentially different.
Place $8$ blue bottles.
Then for the couple of green bottles there are $9$ spots.
After that there are $8$ spots left for the remaining $5$ green bottles.
Result: $$9\binom85=504$$possibilities.