different ways of arranging... bottles side-by-side?

[wind]

How many different ways are there of arranging 7 green and 8 brown bottles in a row so the exactly one pair of green bottles is side-by-side?

Total ways of arranging the shoes
15!/(7!*8!)= 6435
but that does not account for one pair being side by side

thanks

 

[pka]

Are we to assume that the bottles are all distinct?
Or are the green bottles are identical as are the brown bottles?

 

[soroban]

Hello, wind!

How many different ways are there of arranging 7 green and 8 brown bottles
in a row so the exactly one pair of green bottles is side-by-side?

Duct-tape two green bottles together.
Arrange this pair and the 8 brown bottles in a row.
. . There are 9 possible arrangements.

For each of these arrangements,
. . consider the "spaces" between and around the brown bottles.

. .
B
B
B [GG] B
B
B
B
B


The remaining 5 green bottles will be placed in 5 of the 8 spaces.
. . There are: \(\displaystyle {8 \choose 5} \,=\,\)56 ways.


Answer: \(\displaystyle \:9\,\times\,56\:=\:\)504 ways.

 

[wind]

thanks but what does \(\displaystyle {8 \choose 5} \\) mean? I have never seen that before.... :?

and why are there 9 possible arrangements


thanks

 

[pka]

Did you bother to look up “binominal coefficients” on the web?
That is what \(\displaystyle \L {N \choose k} = \frac{{N!}}{{k!\left( {N - k} \right)!}}\).

 

[Denis]

thanks but what does \(\displaystyle {8 \choose 5} \\) mean?

means number of ways to choose 5 things from 8 things!

and why are there 9 possible arrangements

if you don't understand that after Soroban's explanation, then you're not
ready for this kind of problem.

 

[wind]

and why are there 9 possible arrangements

if you don't understand that after Soroban's explanation, then you're not
ready for this kind of problem.

I guess I was not reading properly I understand that now



Is there another way to do this problem without using binomial coefficients, since we have not learned that yet...


thanks