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I have no idea how to begin the following problem

Nautical flags are specially designed flags made up of several colors which can be used to signal from ship to ship, or ship to shore. Suppose there are 4 red, 5 blue and 8 yellow flags. How many different arrangements can be made if all the flags must be used on a vertical flag pole?

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    It's a trick question -- there is no solid blue [maritime signaling](http://en.wikipedia.org/wiki/International_maritime_signal_flags) flag.2012-11-12

3 Answers 3

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Hint Presumably there are $17$ "positions" for the flags. The positions of the red ones can be chosen in $\dbinom{17}{4}$ ways. For each of these ways, the positions of the blue flags can be chosen in $\dots$.

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Hint: "multinomial coefficient"

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There are a total of seventeen slots on the flagpole. The red flags can be placed in $\binom{17}{4}$ ways (we use combinations since the red flags are indistinguishable from each other). Next, we can place the five blue flags in $\binom{13}{5}$ ways (since there are only thirteen remaining spots). Finally, place the eight yellow flags in $\binom{8}{8}$ ways (which is just one - our only option is to fill the remaining eights spots with the eight yellow flags. Thus, there are $ \binom{17}{4}\binom{13}{5} $ distinct placements of the flags.

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    13 Choose 5 ways?2012-11-12