Raising a flag sequence with two colors

Question

A signal pole will hold 6 flags stacked top to bottom: 4 identical red flags and 2 identical blue flags. Two stacks are the same if the color pattern from top to bottom matches. How many distinct color patterns are possible?

Accepted Answer

With only two colors, an arrangement is fully determined once you decide which of the 6 positions hold the blue flags; the reds fill the rest. Using the repeated-items formula: \frac{6!}{4!\,2!} = \frac{720}{24 	imes 2} = \frac{720}{48} = 15. Equivalently, choose the 2 blue positions out of 6: \binom{6}{2} = 15. With exactly two categories, the multiset-arrangement count collapses to a single combination. Takeaway. Arranging items of only two kinds, \dfrac{n!}{a!\,b!}, is the same as \binom{n}{a}. Four red and two blue give 15 patterns.

Raising a flag sequence with two colors

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