Math 2631 Homework 1
due Tuesday, September 5th at 11:59 pm
Instructions: You must compose your assignments independently; however, you may discuss
your work with one another at the rough level.
1. In how many ways can 4 different science fiction books, 3 different math books and 1 cooking
book be arranged in a line, if
(a) (4 points) there are no restrictions on the arrangement?
(b) (4 points) all the science fiction boos have to be stored together and also all the math
books have to be stored together?
2. How many “words” can you create from the letters “ARRANGE”, if
(a) (4 points) you have to use all the letters?
(b) (4 points) you can only use two letters?
3. A tourist wants to visit six out of twelve American cities: seven of them are on the East Coast,
three on the West Coast and two in the middle of the country. In how many ways can the
tourist plan the travel if
(a) (3 points) the order of the visits does not play a role;
(b) (3 points) the order of the visits does matter;
(c) (4 points) the order is not important, but the tourist wants to visit at least three cities
on the East Coast and at least two on the West Coast.
4. (4 points) (A slight deviation from the multiplication rule...) Two experiments are to be
performed in a sequence. The first one can result in r different outcomes (numbers): 1, 2, . . . , r.
If the first experiment results in outcome j (j = 1, 2, . . . , r), then the second experiment can
result in nj possible outcomes. What is the total number of possibles pairs of outcomes of the
two experiments?
n−1
5. Show that Ckn = Ckn−1 + Ck−1
for 1 ≤ k ≤ n − 1. You can prove directly from the definition
of combination or by coming up with a counting example that can be done in two different
ways, corresponding to each side of the equation.
optional problems in the book: Section 1.3 - 32, 33(a)(b)(c), 34, 38(a)(b), 39(a)(b)(c)
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