 11.1.1: A restaurant offers eight appetizers and ten main courses. In how m...
 11.1.2: The model of the car you are thinking of buying is available in nin...
 11.1.3: A popular brand of pen is available in three colors (red, green, or...
 11.1.4: In how many ways can a casting director choose a female lead and a ...
 11.1.5: A student is planning a twopart trip. The first leg of the trip is...
 11.1.6: For a temporary job between semesters, you are painting the parking...
 11.1.7: An ice cream store sells two drinks (sodas or milk shakes), in four...
 11.1.8: A pizza can be ordered with three choices of size (small, medium, o...
 11.1.9: A restaurant offers the following limited lunch menu. Main Course V...
 11.1.10: An apartment complex offers apartments with four different options,...
 11.1.11: Shoppers in a large shopping mall are categorized as male or female...
 11.1.12: There are three highways from city A to city B, two highways from c...
 11.1.13: A person can order a new car with a choice of six possible colors, ...
 11.1.14: A car model comes in nine colors, with or without air conditioning,...
 11.1.15: You are taking a multiplechoice that has five questions. Each of t...
 11.1.16: You are taking a multiplechoice that has eight questions. Each of ...
 11.1.17: In the original plan for area codes in 1945, the first digit could ...
 11.1.18: The local sevendigit telephone numbers in Inverness, California, h...
 11.1.19: License plates in a particular state display two letters followed b...
 11.1.20: How many different fourletter radio station call letters can be fo...
 11.1.21: A stock can go up, go down, or stay unchanged. How many possibiliti...
 11.1.22: A social security number contains nine digits, such as 074667795....
 11.1.23: Explain the Fundamental Counting Principle.
 11.1.24: Figure 11.2 on page 689 shows that a tree diagram can be used to fi...
 11.1.25: Write an original problem that can be solved using the Fundamental ...
 11.1.26: Make Sense? In 2629, determine whether each statement makes sense o...
 11.1.27: Make Sense? In 2629, determine whether each statement makes sense o...
 11.1.28: Make Sense? In 2629, determine whether each statement makes sense o...
Solutions for Chapter 11.1: The Fundamental Counting Principle
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter 11.1: The Fundamental Counting Principle
Get Full SolutionsChapter 11.1: The Fundamental Counting Principle includes 28 full stepbystep solutions. Since 28 problems in chapter 11.1: The Fundamental Counting Principle have been answered, more than 64711 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. This expansive textbook survival guide covers the following chapters and their solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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