Transcription of Math 461 Section P1 Quiz 1 Solution
1 Math 461 Section P1 Quiz 1 SolutionJan 24, how many ways can 3 novels, 2 mathematics books and a chemistrybook 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 betogether?(c) the novels must be together, but the other books can be arrangedin any order? Solution :(a) There are 3 + 2 + 1 = 6 books on the shelf, so the numberof arrangements with no restriction is 6! = 720.(b) Since the mathematics books are all together, we can view themas a single bundle of mathematics books. Two mathematics bookscan be put in any order in the bundle, so the number of ways toget such a bundle is 2!. Similarly there are 3! ways to get a bundleof there are 3! ways to put these two bundles together with thechemistry book onto the shelf in any order, so by the multiplicationprinciple, the number of ways is 3!
2 2! 3! = 72.(c) We use the same idea as in (b). There are 3! different bundles ofnovels, and we have to put this bundle and other 3 books onto theshelf. So the total number of ways is 3! (3 + 1)! = many different paths are there from A to B that go through the pointcircled in the following lattice. Note that at each point you can onlymove up or to the :Every path fromAtoBgoing through the circled pointCcanbe broken into two: a path fromAtoCand a path go fromAtoC, we need 2 moves going up and 2 moves going path is determined by the order we arrange these four moves. Toarrange them, we have to choose 2 positions out of 4 to go up (and theother 2 to go right), so the number of different paths is(42)= we have(31)= 3 paths fromCtoB. So by the multiplicationprinciple, the number of paths fromAtoBis 6 3 = 18.
