18. 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 side-by-side?


To find the number of ways to arrange seven green and eight brown bottles in a row so that exactly one pair of green bottles is side-by-side, follow these steps:

### Step-by-Step Solution:

1. **Pair the Green Bottles:** Start by selecting the one pair of green bottles that will be side-by-side. This can be any one of the 6 pairs of green bottles (since there are 7 green bottles total, and we need exactly one pair). Choose this pair in \(6\) ways.

2. **Treat the Pair as a Single Unit:** Once the pair of green bottles is chosen and grouped together, treat this pair as a single "super green" unit. Now, instead of having 7 green bottles, you have 6 green items (1 "super green" unit and 5 individual green bottles).

3. **Count Total Units to Arrange:** Now, we have 14 units in total: 6 green items (including the "super green" pair) and 8 brown bottles.

4. **Arrange the Units:** Arrange these 14 units (6 green units and 8 brown units). The number of ways to do this is given by the binomial coefficient:

   \[
   \text{Total arrangements} = \binom{14}{6}
   \]

   where \( \binom{14}{6} \) represents choosing 6 positions out of 14 for the green units.

5. **Calculate the Binomial Coefficient:**

   \[
   \binom{14}{6} = \frac{14!}{6! \times 8!} = 3003
   \]

6. **Multiply by the Ways to Choose the Green Pair:**
   
   Since the "super green" pair can be any one of the 6 pairs of green bottles:

   \[
   \text{Total ways} = 6 \times 3003 = 18018
   \]

### Final Answer:

There are **18,018** different ways to arrange seven green and eight brown bottles in a row, so that exactly one pair of green bottles is side-by-side.

Would you like further explanation on any step, or do you have more questions? Here are some related queries:

1. What if we wanted two pairs of green bottles to be side-by-side?
2. How would the answer change if we had fewer green bottles?
3. How does this method apply if the bottles were distinguishable by more than just color?
4. Can you explain why we use the binomial coefficient in these arrangements?
5. How would the calculation change if no pairs of green bottles could be side-by-side?

**Tip:** When calculating arrangements with specific conditions, grouping items can simplify the problem and ensure that restrictions are met correctly.

Explore the mathematical problem of arranging seven green and eight brown bottles in a row such that exactly one pair of green bottles is side-by-side. Learn the application of binomial coefficients and combinatorial principles in solving this problem. Suitable for advanced high school students and enthusiasts of combinatorial mathematics.

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