Different flag signal questions

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How many different signals can be created by lining up 9 flags in a vertical column in 3 flags are white, 2 are red, and 4 are blue? Is it 9 choose 3 * 6 choose 2 * 4 choose 4?

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To put it simply, there are $9!$ ways in all, if the same-colored flags are considered distinct. However, they are counted as the same, so we consider how many times is each configuration repeated.

There are 3! ways for the white, 2! for the red, 4! for the blue, so the final answer is $\frac{9!}{3!2!4!}$. This is the same as the quantity you gave above.

Your method is another nice method to solve it.

This is the explanation, for others who are interested. We first have 9 uncolored flags. There are 9choose3 ways to choose 3 of them to make white. so there are 6 left. Now we choose 2 of them to make red (there are 6choose2 ways). So there are 4 left and there is 4choose4, or 1 way to do it.