 9.3.12E: a. How many ways can the letters of the word THEORY be arranged in ...
 9.3.1E: a. How many bit strings consist of from one through four digits? (S...
 9.3.2E: a. How many strings of hexadecimal digits consist of from one throu...
 9.3.3E: a. How many integers from 1 through 999 do not have any repeated di...
 9.3.4E: How many arrangements in a row of no more than three letters can be...
 9.3.5E: a. How many fivedigit integers (integers from 10,000through 99,999...
 9.3.6E: In a certain state, license plates consist of from zero to three le...
 9.3.7E: In another state, all license plates consist of from four to six sy...
 9.3.8E: At a certain company, passwords must be from 3–5 symbols long and c...
 9.3.9E: a. Consider the following algorithm segment: How many times will th...
 9.3.10E: A calculator has an eightdigit display and a decimal point that is...
 9.3.11E: a. How many ways can the letters of the word QUICK be arranged in a...
 9.3.13E: A group of eight people are attending the movies together.a. Two of...
 9.3.14E: An early compiler recognized variable names according to the follow...
 9.3.15E: Identifiers in a certain database language must begin with a letter...
 9.3.16E: a. If any seven digits could be used to form a telephone number, ho...
 9.3.17E: a. How many strings of four hexadecimal digits do not have any repe...
 9.3.18E: Just as the difference rule gives rise to a formula for the probabi...
 9.3.19E: A combination lock requires three selections of numbers, each from ...
 9.3.20E: a. How many integers from 1 through 100,000 contain the digit 6 exa...
 9.3.21E: Six new employees, two of whom are married to each other, are to be...
 9.3.22E: Consider strings of length n over the set {a, b, c, d}.a. How many ...
 9.3.23E: a. How many integers from 1 through 1,000 are multiples of 4 or mul...
 9.3.24E: a. How many integers from 1 through 1,000 are multiples of 2 or mul...
 9.3.25E: Counting Strings:a. Make a list of all bit strings of lengths zero,...
 9.3.26E: Counting Strings: Consider the set of all strings of a’s, b’s, and ...
 9.3.27E: For each integer n ? 0, let ak be the number of bit strings of leng...
 9.3.28E: For each integer n ? 2 let an be the number of permutations of {1, ...
 9.3.29E: Refer to Example 9.3.5.a. Write the following IP address in dotted ...
 9.3.30E: A row in a classroom has n seats. Let sn be the number of ways none...
 9.3.31E: Assume that birthdays are equally likely to occur in any one of the...
 9.3.32E: Assuming that all years have 365 days and all birthdays occur with ...
 9.3.33E: A college conducted a survey to explore the academic interests and ...
 9.3.34E: A study was done to determine the efficacy of three different drugs...
 9.3.35E: An interesting use of the inclusion/exclusion rule is to check surv...
 9.3.36E: Fill in the reasons for each step below. If A and B are sets in a f...
 9.3.37E: For each of exercises below, the number of elements in a certain se...
 9.3.38E: For each of exercises below, the number of elements in a certain se...
 9.3.39E: For each of exercises below, the number of elements in a certain se...
 9.3.40E: For use the definition of the Euler phi functionEuler phi functionF...
 9.3.41E: For use the definition of the Euler phi functionEuler phi functionF...
 9.3.42E: A gambler decides to play successive games of blackjack until he lo...
 9.3.43E: A derangement of the set {1, 2,…, n} is a permutation that moves ev...
 9.3.44E: Note that a product x1 x2x3 may be parenthesized in two different w...
 9.3.45E: Use mathematical induction to prove Theorem.TheoremThe Addition Rul...
 9.3.46E: Prove the inclusion/exclusion rule for two sets A and B by showing ...
 9.3.47E: Prove the inclusion/exclusion rule for three sets.
 9.3.48E: Use mathematical induction to prove the general inclusion/exclusion...
 9.3.49E: A circular disk is cut into n distinct sectors, each shaped like a ...
Solutions for Chapter 9.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 9.3
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions. Since 49 problems in chapter 9.3 have been answered, more than 57082 students have viewed full stepbystep solutions from this chapter. Chapter 9.3 includes 49 full stepbystep solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.