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Math 461 Section P1 Quiz 1 Solution Jan 24, 2013 1. In how many ways can 3 novels, 2 mathematics books and a chemistry book be arranged on a shelf if: (a) the books can be arranged in any order? (b) the mathematics books must be together and the novels must be together? (c) the novels must be together, but the other books can be arranged in any order? Solution: (a) There are 3 + 2 + 1 = 6 books on the shelf, so the number of arrangements with no restriction is 6! = 720. (b) Since the mathematics books are all together, we can view them as a single bundle of mathematics books. Two mathematics books can be put in any order in the bundle, so the number of ways to get such a bundle is 2!. Similarly there are 3! ways to get a bundle of novels. Now there are 3! ways to put these two bundles together with the chemistry book onto the shelf in any order, so by the multiplication principle, the number of ways is 3! × 2! × 3! = 72. (c) We use the same idea as in (b). There are 3! different bundles of novels, and we have to put this bundle and other 3 books onto the shelf. So the total number of ways is 3! × (3 + 1)! = 144. 2. How many different paths are there from A to B that go through the point circled in the following lattice. Note that at each point you can only move up or to the right. C Solution: Every path from A to B going through the circled point C can be broken into two: a path from A to C and a path from C to B. To go from A to C, we need 2 moves going up and 2 moves going right. The path is determined by the order we arrange these four moves. To arrange them, we have to choose 2 positions out of 4 to go up (and the other 2 to go right), so the number of different paths is 42 = 6. Similarly we have 31 = 3 paths from C to B. So by the multiplication principle, the number of paths from A to B is 6 × 3 = 18.