[Math] How many different vertical arrangements are there of 10 flags if…

Question

How many different vertical arrangements are there of 10 flags if 4 are white, 3 are blue,
2 are green and 1 is red?

I know the answer is 12 600 but am not sure how to get to it. Could someone walk me through this please?

Answer 1

This is the solution.

Suppose that all of the flags (even same-colored), are distinct. Then there are 10! ways.

Now we count how many times each arrangement is repeated since same-colored flags are considered the same. 4 white flags could be done in 4! ways (if they are distinguishable). Similar for the other colors. Thus # of repetitions is $4!\cdot3!\cdot2!\cdot1!$.

Thus we divide 10! by the number of repetitions to get $\frac{10!}{4!\cdot 3!\cdot 2!\cdot 1!}=12600$.