 6.5.1E: In how many different ways can five elements be selected in order f...
 6.5.3E: In how many different ways can five elements be selected in order f...
 6.5.2E: In how many different ways can five elements be selected in order f...
 6.5.4E: Every day a student randomly chooses a sandwich for lunch from a pi...
 6.5.5E: How many ways are there to assign three jobs to five employees if e...
 6.5.6E: How many ways are there to select five unordered elements from a se...
 6.5.7E: How many ways are there to select three unordered elements from a s...
 6.5.8E: How many different ways are there to choose a dozen donuts from the...
 6.5.10E: A croissant shop has plain croissants, cherry croissants, chocolate...
 6.5.12E: How many different combinations of pennies, nickels, dimes, quarter...
 6.5.9E: A bagel shop has onion bagels, poppy seed bagels, egg bagels, salty...
 6.5.11E: How many ways are there to choose eight coins from a piggy bank con...
 6.5.13E: A book publisher has 3000 copies of a discrete mathematics book. Ho...
 6.5.14E: How many solutions are there to the equationx1+ x2 + x3 + x4 = 17,w...
 6.5.15E: How many solutions are there to the equationx1+ x2 + x3 + x4+ x5 = ...
 6.5.16E: How many solutions arc there to the equationx1+ x2 + x3 + x4+ x5+ x...
 6.5.17E: How many strings of 10 ternary digits (0, 1, or 2) are there that c...
 6.5.18E: How many strings of 20decimal digits are there that contain two 0s...
 6.5.20E: How many solutions are there to the inequalityx1+ x2 + x3 = 11,wher...
 6.5.19E: Suppose that a large family has 14 children, including two sets of ...
 6.5.21E: How many ways are there to distribute six indistinguishable balls i...
 6.5.22E: How many ways are there to distribute 12 indistinguishable balls in...
 6.5.24E: How many ways are there to distribute 15 distinguishable objects in...
 6.5.23E: How many ways are there to distribute 12 distinguishable objects in...
 6.5.25E: How many positive integers less than 1,000,000 have the sum of thei...
 6.5.27E: There are 10 questions on a discrete mathematics final exam. How ma...
 6.5.26E: How many positive integers less than 1,000,000 have exactly one dig...
 6.5.28E: Show that there are C(n + r – q1 – q2 – ? ? qr? 1, n ? q1 – q2 – ? ...
 6.5.29E: How many different bit strings can be transmitted if the string mus...
 6.5.32E: How many different strings can be made from the letters in AARDVARK...
 6.5.31E: How many different strings can be made from the letters in ABRACADA...
 6.5.33E: How many different strings can be made from the letters in ORONO, u...
 6.5.34E: How many strings with five or more characters can be formed from th...
 6.5.35E: How many strings with seven or more characters can be formed from t...
 6.5.37E: A student has three mangos, two papayas, and two kiwi fruits. If th...
 6.5.36E: How many different bit strings can be formed using six 1s and eight...
 6.5.39E: How many ways are there to travel in xyz space from the origin (0, ...
 6.5.41E: How many ways are there to deal hands of seven cards to each of fiv...
 6.5.38E: A professor packs her collection of 40 issues of a mathematics jour...
 6.5.42E: In bridge, the 52 cards of a standard deck are dealt to four player...
 6.5.40E: How many ways are there to travel in xyzw space from the origin (0,...
 6.5.43E: How many ways are there to deal hands of five cards to each of six ...
 6.5.44E: In how many ways can a dozen books be placed on four distinguishabl...
 6.5.47E: Use the product rule to prove Theorem 4, by first placing objects i...
 6.5.48E: Prove Theorem 4 by first setting up a onetoone correspondence bet...
 6.5.46E: A shelf holds 12 books in a row. How many ways are there to choose ...
 6.5.49E: In this exercise we will prove Theorem 2 by setting up a onetoone...
 6.5.45E: How many ways can n books be placed on k distinguishable shelvesa) ...
 6.5.52E: How many ways are there to put five temporary employees into four i...
 6.5.51E: How many ways are there to distribute six distinguishable objects i...
 6.5.50E: How many ways are there to distribute five distinguishable objects ...
 6.5.53E: How many ways are there to put six temporary employees into four id...
 6.5.55E: How many ways are there to distribute six indistinguishable objects...
 6.5.54E: How many ways are there to distribute five indistinguishable object...
 6.5.56E: How many ways are there to pack eight identical DVDs into five indi...
 6.5.57E: How many ways are there to pack nine identical DVDs into three indi...
 6.5.58E: How many ways are there to distribute five balls into three boxes i...
 6.5.60E: Suppose that a basketball league has 32 teams, split into two confe...
 6.5.62E: How many different terms are there in the expansion of (x1 + x2 +?+...
 6.5.61E: Suppose that a weapons inspector must inspect each of five differen...
 6.5.64E: Find the expansion of (x + y + z)4.
 6.5.63E: Prove the Multinomial Theorem: If n is a positive integer, then whe...
 6.5.65E: Find the coefficient of x3y2z5 in (x + y + z)10.
 6.5.66E: How many terms are there in the expansion of (x + y + z)100?
Solutions for Chapter 6.5: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 6.5
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.5 includes 64 full stepbystep solutions. Since 64 problems in chapter 6.5 have been answered, more than 220007 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.