Number of Ways in Which 7 Green Bottles and 8 Blue Bottles can be Arranged GMAT Problem Solving

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 alike except for the color).

1. 84
2. 360
3. 504
4. 512
5. 575

Answer:
Solution with Explanation:
Approach Solution (1):

There are 8 identical blue bottles and 7 identical green bottles. We want that 2 green bottles be together and so none of the remaining 5 green bottles can be together.
Let us take the 2 green bottles as 1 unit.
Let us first arrange the Blue bottles with the number of spaces between them indicated by numbers as shown below. In these spaces, we will first put the pair of green bottles and then place the remaining 5 bottles in the remaining spaces.
1 B1 2 B2 3 B3 4 B4 5 B5 6 B6 7 B7 8 B8 9
Since the 8 blue bottles are identical, there is only 1 way of arranging them.
Now for the pair of green bottles, there are 9 places for them to choose. The number of ways = 9C1 = 9 ways
For the last set of 5 bottles, they have 8 places to choose from = 8C5 = 8C3 =\(\frac{8*7*6}{3*2}\)= 56 ways
Therefore the total number of ways = 1 * 9 * 56 = 504

Correct Option: C

“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 alike except for the color).”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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