Quiz 1 and solutions

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
(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.
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.