Created by StemPy LabFour married couples, eight people in all, sit at a round table with eight equal chairs. Two seatings count as the same when one is a rotation of the other, but a seating and its mirror image count as different. In how many seatings does no husband sit next to his own wife?
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← The figure shows the graph of a function f on [0, 12]. It is made of three straight pieces joining the points (0, 0), (3, 6), (9, 0) and (12, -3): a steep rise, a slow fall, and a dip below the axis. For 0 ≤ x ≤ 9 let F(x) = ∫xˣ⁺³ f(t) dt, the area under the graph over a window of width 3 that slides along the axis. The window catches the most area when its two ends are at the same height, but the peak of f need not sit in the middle of the window. What is the greatest value of F(x) for 0 ≤ x ≤ 9?The figure shows a regular polygon with 12 vertices, all on one circle. Any three of the vertices form a triangle, which gives C(12, 3) = 220 triangles in all. How many of these triangles are acute-angled, that is, have all three angles less than 90°? →
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