 13.1.1: 0! = ; 1! = . (p. 801)
 13.1.2: True or False n! = 1n + 12! n . (p. 801)
 13.1.3: A(n) is an ordered arrangement of r objects chosen from n objects.
 13.1.4: A(n) is an arrangement of r objects chosen from n distinct objects,...
 13.1.5: P1n, r2 =
 13.1.6: C1n, r2 =
 13.1.7: In 714, find the value of each permutation
 13.1.8: In 714, find the value of each permutation
 13.1.9: In 714, find the value of each permutation
 13.1.10: In 714, find the value of each permutation
 13.1.11: In 714, find the value of each permutation
 13.1.12: In 714, find the value of each permutation
 13.1.13: In 714, find the value of each permutation
 13.1.14: In 714, find the value of each permutation
 13.1.15: In 1522, use formula (2) to find the value of each combination.
 13.1.16: In 1522, use formula (2) to find the value of each combination.
 13.1.17: In 1522, use formula (2) to find the value of each combination.
 13.1.18: In 1522, use formula (2) to find the value of each combination.
 13.1.19: In 1522, use formula (2) to find the value of each combination.
 13.1.20: In 1522, use formula (2) to find the value of each combination.
 13.1.21: In 1522, use formula (2) to find the value of each combination.
 13.1.22: In 1522, use formula (2) to find the value of each combination.
 13.1.23: List all the ordered arrangements of 5 objects a, b, c, d, and e ch...
 13.1.24: List all the ordered arrangements of 5 objects a, b, c, d, and e ch...
 13.1.25: List all the ordered arrangements of 4 objects 1, 2, 3, and 4 choos...
 13.1.26: List all the ordered arrangements of 6 objects 1, 2, 3, 4, 5, and 6...
 13.1.27: List all the combinations of 5 objects a, b, c, d, and e taken 3 at...
 13.1.28: List all the combinations of 5 objects a, b, c, d, and e taken 2 at...
 13.1.29: List all the combinations of 4 objects 1, 2, 3, and 4 taken 3 at a ...
 13.1.30: List all the combinations of 6 objects 1, 2, 3, 4, 5, and 6 taken 3...
 13.1.31: How many twoletter codes can be formed using the letters A, B, C, ...
 13.1.32: List all the combinations of 6 objects 1, 2, 3, 4, 5, and 6 taken 3...
 13.1.33: How many threedigit numbers can be formed using the digits 0, 1, 2...
 13.1.34: How many threedigit numbers can be formed using the digits 0, 1, 2...
 13.1.35: In how many ways can 4 people be lined up?
 13.1.36: In how many ways can 5 different boxes be stacked?
 13.1.37: How many different threeletter codes are there if only the letters...
 13.1.38: How many different fourletter codes are there if only the letters ...
 13.1.39: Companies whose stocks are listed on the New York Stock Exchange (N...
 13.1.40: Stocks on the NASDAQ Companies whose stocks are listed on the NASDA...
 13.1.41: In how many ways can a committee of 4 students be formed from a poo...
 13.1.42: In how many ways can a committee of 3 professors be formed from a d...
 13.1.43: How many arrangements of answers are possible for a true/false test...
 13.1.44: How many arrangements of answers are possible in a multiplechoice ...
 13.1.45: Five different mathematics books are to be arranged on a students d...
 13.1.46: How many different license plate numbers can be made using 2 letter...
 13.1.47: In how many ways can 2 people each have different birthdays? Assume...
 13.1.48: In how many ways can 5 people each have different birthdays? Assume...
 13.1.49: A student dance committee is to be formed consisting of 2 boys and ...
 13.1.50: The student relations committee of a college consists of 2 administ...
 13.1.51: How many different 9letter words (real or imaginary) can be formed...
 13.1.52: How many different 11letter words (real or imaginary) can be forme...
 13.1.53: An urn contains 7 white balls and 3 red balls. Three balls are sele...
 13.1.54: An urn contains 15 red balls and 10 white balls. Five balls are sel...
 13.1.55: The U.S. Senate has 100 members. Suppose that it is desired to plac...
 13.1.56: A defensive football squad consists of 25 players. Of these, 10 are...
 13.1.57: In the American Baseball League, a designated hitter may be used to...
 13.1.58: In the National Baseball League, the pitcher usually bats ninth. If...
 13.1.59: A baseball team has 15 members. Four of the players are pitchers, a...
 13.1.60: In the World Series the American League team 1A2 and the National L...
 13.1.61: A basketball team has 6 players who play guard (2 of 5 starting pos...
 13.1.62: On a basketball team of 12 players, 2 only play center, 3 only play...
 13.1.63: A combination lock displays 50 numbers. To open it, you turn clockw...
 13.1.64: Make up a problem different from any found in the text that require...
 13.1.65: Make up a problem different from any found in the text that require...
 13.1.66: Explain the difference between a permutation and a combination. Giv...
Solutions for Chapter 13.1: Sequences; Induction; the Binomial Theorem
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 13.1: Sequences; Induction; the Binomial Theorem
Get Full SolutionsChapter 13.1: Sequences; Induction; the Binomial Theorem includes 66 full stepbystep solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Since 66 problems in chapter 13.1: Sequences; Induction; the Binomial Theorem have been answered, more than 59555 students have viewed full stepbystep solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. This expansive textbook survival guide covers the following chapters and their solutions.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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

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.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).