Discrete Mathematics for Computer Science

Chapter Review 471

THEOREMS

Binomial Theorem Pascal's Identity
Newton's Identity Sums of Powers

7.12.2 Starting to Review

1. What are the values of C(5, 3) and P(5, 3)?
   (a) 60, 10
   (b) 60, 20
   (c) 60
   (d) None of the above


2. There are 12 roads to Merced, 8 roads from Merced to Planter, and 13 roads from
   Planter to San Francisco. How many possible ways are there to get to San Francisco
   from Merced?


3. How many ways can you roll two dice and get a total of six appearing on the top faces?


4. Names on Ork are formed according to the rules that a name is six characters long and
   has two vowels. The vowels may not be in either the first or the last positions, and the
   two vowels may not occur in adjacent locations. How many people can live on Ork
   with each person having a different name?


5. To graduate, Sally needs two courses to complete the general education requirement.
   Courses in anthropology and economics will satisfy the requirement. If Sally has sat-
   isfied the prerequisites for 18 anthropology and 21 economics courses, how many
   schedules are possible?


6. A bakery sells six different kinds of pastries. How many different dozens of pastry can
   you buy? What if you buy at least one of each kind?


7. There are 10 geography books, 12 chemistry books, and 18 detective novels. How
   many ways can you pick two books from each of two different groups of books?


8. Using the Binomial Theorem, compute 114 .Show all work.

9. Find n such that C(n, 0) + C(n, 1) + ... + C(n, n) = 128.

10. Compute the coefficient of x^3 y^2 zw^2 in the expansion of (x - y +^2 z - 2w)^8.

712.3 Review Questions

1. Find the number of ways in which nine 3's and six 5's can be placed in a row so that
   no two 5's are together.


2. Determine the number of sequences of length r for any r E N if the first portion is
   comprised of the letters a, b, c, and d with the remainder being comprised of the Greek
   letters {a, fi, y, )}.


3. In the interest of efficiency, spelling rules for words have been revised. The word relief
   can be spelled in the ways described by the following rules:
   " The number of letters must not exceed six.
   " The word must contain at least one 1.
   "* The word must begin with an r and end with anf

"* There is just one r and onef and only the letters e, i, and 1 may occur in the middle

positions.

How many ways can relief be spelled?