A flagpole has spaces for seven colored flags arranged in a vertical line. Two of the flags are yellow, two are green, one is red, one is orange, and one is brown. Flags are to be placed on the pole under the following conditions: The orange flag is to be placed immediately below the brown flag. The red flag is not allowed to be immediately above or below the green flag. The two green flags must be together. The two yellow flags must not be together. The red flag is not allowed to be at the top or the bottom of the pole. How many acceptable flag arrangements are there?
2026-03-29 15:53:24.1774799604
Acceptable Arrangements
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(This is not a problem about finding and applying the correct formula, but a problem about organizing the diversity of cases in a way that minimizes the number of "thought steps".)
Arrange the flags horizontally from left to right.
Brown followed by orange form a single flag B, and the two green flags form a single flag G.
So we are left with five flags y, y, r, B, G to be arranged on spots $1$, $2$, $3$, $4$, $5$.
When r is at spot 3, G may be on spot $1$ or $5$, but B cannot be on the same side of r (or the two y would be adjacent). In any case there are $2$ choices left for B, so there are $4$ possibilities in all.
When r is at spot $2$ then G can be at spot $4$ or $5$. When G is at spot $4$ then B can be at any of the three empty spots $1$, $3$, $5$; and when G is at spot $5$ then B can be at $3$ or $4$, but not at $1$. It follows that there are $5$ possibilities in this case, and by symmetry there are another $5$ possibilities when r is at spot $4$.
Collecting it all we can say that there are $14$ acceptable flag arrangements.