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Number of Arrangements?

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asa.hoshi:
Calculate the number of arrangements of 2 red, 3 green and 4 blue bottles in a line, given that at least 2 bottles of the same colour are always to be in succession.

I know that...
#arrangements w/o restrictions is ways

need to find # arrangements so at least 2 bottles of the same colour are in succession.
So,

if i take #arrangements w/o restrictions - #arrangements with no colour in succession,

would that work out? but im having heaps of toruble to count #arrangements with no colour in succession... hrm, is there another way to approach this question?

kamil9876:
Yes, that's a good start.

To solve our next little sub-problem, I'll show you that it is easier if we assume there are only two kinds of bottles, ie let us forget about the red.

So we have:

_ B _ B _ B _ B _

and we must choose exactly 3 of the _ to place our G.

So there are different ways of doing this.

Now how do I also include the fact that there are 2 red bottles?

Well for any arrangement of the 7 non-red bottles looks like this:

_ X _ X _ X _ X _ X _ X _ X _ X _

Where X denotes any of the non-red bottles. We know how many ways there are of arranging the X's, and now we know that for each arrangement we must choose exactly 2 of the _ to place the red bottles. There are ways of doing this.

So in total there are ways.

asa.hoshi:
Thanks. but am i see something, but does this way allows GBGBGBB when you shouldn't (in the 1st section)?

kamil9876:
actually yeah fail.

asa.hoshi:

--- Quote from: kamil9876 on October 04, 2010, 12:37:53 am ---actually yeah fail.

--- End quote ---
u gave a better attempt than i did! HAHA.

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