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

W

wind

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Sep 20, 2006
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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
 
Are we to assume that the bottles are all distinct?
Or are the green bottles are identical as are the brown bottles?
 
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?
Click to expand...

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.

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

and why are there 9 possible arrangements


thanks
 
Did you bother to look up “binominal coefficients” on the web?
That is what \(\displaystyle \L {N \choose k} = \frac{{N!}}{{k!\left( {N - k} \right)!}}\).
 
wind said:
thanks but what does \(\displaystyle {8 \choose 5} \\) mean?
Click to expand...
means number of ways to choose 5 things from 8 things!

wind said:
and why are there 9 possible arrangements
Click to expand...
if you don't understand that after Soroban's explanation, then you're not
ready for this kind of problem.
 
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.
Click to expand...

I guess I was not reading properly :oops: I understand that now



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


thanks
 
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