How many different ways are there of arranging seven green and eight brown bottles in a row, so that exactly one pair of green bottles is adjacent?

To find the number of different ways to arrange seven green and eight brown bottles in a row such that exactly one pair of green bottles is adjacent, we can follow these steps:

### Step 1: Arrange the Green Bottles
1. **Pairing One Green Bottle**: Choose 2 out of the 7 green bottles to be the pair that will be adjacent. The number of ways to do this is given by the combination:
   \[
   \text{Number of ways to choose the pair} = \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21
   \]

2. **Treat the Pair as a Single Unit**: After selecting the pair, treat it as a single unit. Now, you have 6 individual green bottles and 1 pair (which acts as 1 unit), making a total of 7 units.

### Step 2: Arrange the Units and the Brown Bottles
1. **Arrange the 7 Units**: The number of ways to arrange the 7 units (6 individual green bottles and 1 pair) is \(7!\).

2. **Insert the 8 Brown Bottles**: Now, consider the positions between and around these 7 units where the 8 brown bottles can be placed. There are 8 positions (before the first unit, between the units, and after the last unit). The number of ways to arrange the 8 brown bottles in these 8 positions is given by:
   \[
   \text{Number of ways to arrange the brown bottles} = \binom{14}{8} = \frac{14!}{8! \times 6!}
   \]

### Step 3: Calculate the Total Number of Arrangements
Multiply the number of ways to choose the pair, arrange the units, and place the brown bottles:

\[
\text{Total Number of Arrangements} = \binom{7}{2} \times 7! \times \binom{14}{8}
\]

Let's compute each part:

\[
\binom{14}{8} = \frac{14!}{8! \times 6!} = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 3003
\]

\[
7! = 5040
\]

Now, the total number of arrangements:

\[
\text{Total Number of Arrangements} = 21 \times 5040 \times 3003
\]

Let's calculate this:

\[
21 \times 5040 = 105840
\]
\[
105840 \times 3003 = 317965920
\]

So, the total number of different ways to arrange the seven green and eight brown bottles in a row, with exactly one pair of green bottles adjacent, is **317,965,920**.

Would you like more details on any part of the solution, or do you have any questions?

### Related Questions:
1. How would the answer change if no green bottles are adjacent?
2. What if we have a different number of brown bottles, say 7, instead of 8?
3. What if there are two pairs of adjacent green bottles instead of one?
4. How would the problem change if the bottles were indistinguishable by color?
5. What if we want at least one pair of green bottles adjacent instead of exactly one pair?

### Tip:
When dealing with problems involving arrangements and restrictions, breaking the problem into smaller, manageable parts (like grouping units and considering combinations) can greatly simplify the solution process.

Discover how to calculate the number of ways to arrange seven green and eight brown bottles in a row with exactly one pair of green bottles adjacent. Detailed step-by-step solution including permutation and combination formulas.

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