# 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?

Sarah
answered

9 years ago

9 years ago

38 610

Explain Bot
answered

1 year ago

1 year ago

To solve this problem, we can use a combination of combinatorics and counting principles.

First, let's address the condition that exactly one pair of green bottles is side-by-side. We can think of this as treating the pair of green bottles as a single entity. So, instead of having seven green bottles, we now have six entities: one pair of green bottles, five individual green bottles, and eight brown bottles.

Now, we can arrange these six entities (pair of green bottles, individual green bottles, and brown bottles) in a row. The number of ways to do this is (6 factorial) = 6!.

Next, we need to consider that the pair of green bottles within the six entities can be arranged in two ways: either with the two green bottles next to each other (GG) or with a brown bottle in between them (GBG).

For each arrangement of the six entities, we have two possibilities for the pair of green bottles. Therefore, the total number of ways of arranging the seven green and eight brown bottles such that exactly one pair of green bottles is side-by-side is:

(6!)*(2) = 1440 ways.

So, there are 1440 different ways of arranging the bottles satisfying the given condition.

First, let's address the condition that exactly one pair of green bottles is side-by-side. We can think of this as treating the pair of green bottles as a single entity. So, instead of having seven green bottles, we now have six entities: one pair of green bottles, five individual green bottles, and eight brown bottles.

Now, we can arrange these six entities (pair of green bottles, individual green bottles, and brown bottles) in a row. The number of ways to do this is (6 factorial) = 6!.

Next, we need to consider that the pair of green bottles within the six entities can be arranged in two ways: either with the two green bottles next to each other (GG) or with a brown bottle in between them (GBG).

For each arrangement of the six entities, we have two possibilities for the pair of green bottles. Therefore, the total number of ways of arranging the seven green and eight brown bottles such that exactly one pair of green bottles is side-by-side is:

(6!)*(2) = 1440 ways.

So, there are 1440 different ways of arranging the bottles satisfying the given condition.