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Combinatorics, Math 145 Homework one
1. Problems from section 1.8: 8, 24, 31, 34.
2. You are on vacation and wish to send your n most favorite professors 2 different postcards each. There are k kinds of postcards. How many different ways are there to do this? (note: it is ok if two professors get the same card, after all they are from different departments).
3. Draw n lines in the plane in such way that no two are parallel and no three intersect in a common point. Prove that the plane is divided into n(n+1) 2 + 1 regions.
4. Among the integer numbers 1, 2, . . . 1010, are there more of those containing 9 in their decimal notation or those with no 9?
5. How many permutations of 1, 2, . . . n have a single cycle?
6. Device a way to compute the order of a permutation. Test your algorithm with the permutation [2, 3, 1, 5, 4, 7, 8, 9, 6].
7. Consider the numbers 1, 2, . . . , 1000, show that among any 501 of them there are two numbers such that one divides the other one. Hint: think of a way to use the pigeonhole principle.