MATH 681 Problem Set #5 This problem set is due at the beginning of class on November 10 . 1. (5 points) Find an asymptotically accurate approximation for ( n 2 n ) in terms of poly-nomials, exponentials, and self-exponentials. You may write it in big-O notation if you wish. 2. (10 points) You have a large supply of beads of 4 diﬀerent colors and want to string eight of them on a necklace, making use of each bead at least once . How many ways are there to do so, if necklaces are considered identical if they are rotations or reﬂections of each other? 3. (25 points) Answer the following questions about icosohedron-coloring. You may ﬁnd it useful to assemble the model at http://www.korthalsaltes.com/model.php?name_en=icosahedron in order to help your visualization. (a) (5 points) Identify the 60 rotation-permutations of the icosohedron. You need not explicitly give all 60; merely give a classiﬁcation scheme which identiﬁes 60 diﬀerent rotations.
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