 8.1.1: On a particular afternoon, a student plans to take her niece to a m...
 8.1.2: Draw a tree diagram for Example 8.1.
 8.1.3: Each seat in an arena is labeled with a letter of the alphabet foll...
 8.1.4: A new automobile can be ordered with 8 different exterior colors an...
 8.1.5: At a certain restaurant, a dinner starts with soup, salad or juice,...
 8.1.6: Suppose that a quiz in a discrete mathematics course consists of 15...
 8.1.7: Let A and B be two sets with A = 5 and B = 6. (a) How many diff...
 8.1.8: Each student at a certain university is given a 6digit code (such ...
 8.1.9: In how many orders can 4 married couples be seated in a row of 8 ch...
 8.1.10: How many different 3digit numbers can be formed using the digits 1...
 8.1.11: How many different 4digit numbers can be formed from the digits 1,...
 8.1.12: How many 4digit numbers can be formed from the digits 1, 2, . . . ...
 8.1.13: A student is preparing to go to class on a winter morning. He has t...
 8.1.14: A man leaves for work on a rainy morning. He has a choice of three ...
 8.1.15: It is decided to have dinner at a Chinese, Italian or Mexican resta...
 8.1.16: If radio station call letters consist of either 3 or 4 letters and ...
 8.1.17: At a certain university, a telephone number begins with 355 or 357 ...
 8.1.18: How many different 8bit strings begin with 010 and end with 11?
 8.1.19: How many different 10bit strings begin and end with the same 5bit...
 8.1.20: How many different 7bit strings (a) begin with 1011 and end with 1...
 8.1.21: How many different 7bit strings begin with 11 or 0011?
 8.1.22: How many different 10bit strings begin with 1011 or 0110?
 8.1.23: How many 10bit strings begin with 1011 or 001100?
 8.1.24: How many different 8bit strings end with 011 or end with 01011?
 8.1.25: A license plate consists of a sequence of three letters followed by...
 8.1.26: A password on a computer system consists of four characters, each o...
 8.1.27: A faculty committee has decided to choose one or more students to j...
 8.1.28: Give an example of a counting problem whose answer is 15 44. (Possi...
Solutions for Chapter 8.1: The Multiplication and Addition Principles
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 8.1: The Multiplication and Addition Principles
Get Full SolutionsChapter 8.1: The Multiplication and Addition Principles includes 28 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 28 problems in chapter 8.1: The Multiplication and Addition Principles have been answered, more than 13076 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

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.

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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.