05 Advanced Problem Solving Combinatorics 2 - IMSA
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05 Advanced Problem Solving Combinatorics 2 - IMSA
1) Find the number of arrangements of 111222333 that do not contain three
consecutive identical digits. Show your work!
Advanced Problem Solving Combinatorics II Fall 2013
Name: ________________________
1) Find the number of arrangements of 111222333 that do not contain three consecutive identical digits. Show your work!
2) a) By constructing a generating function, determine the number of ways there are to write 30 as a sum of primes. (Note: 1 is NOT prime!!) b) Repeat to determine the number of ways to write 30 as a sum of distinct primes.
3) Given a recurrence relations an+1 = Can + Dan-1. If one solutions is an = 3n and another solutions is an = (–2)n, then determine the coefficients C and D and determine that value of a12 is the sequence begins a1 = 1, a2 = 2.
4) Determine the number of indistinguishable ways of painting the faces of a cube with three different colors. (Note: 36 = 729; you should find 24 different rotations of the cube. This is because you can pick any one of six sides to face up, and then rotate the cube to any of four positions. These 24 rotations can be divided into three classes: rotations of 90°, 180°, or 270° around the three axes through centers of opposite pairs of faces (9 total rotations), rotations of 120° or 240° around the four axes through opposite vertices (8 total rotations), and 180° rotations about axes through the midpoints of opposite edges (6 rotations). That totals 23—and don’t forget the identity transformation!)
