 11.6.1: The ________ ________ ________ states that if there are ways for on...
 11.6.2: An ordering of elements is called a ________ of the elements.
 11.6.3: The number of permutations of elements taken at a time is given by ...
 11.6.4: The number of ________ ________ of objects is given by
 11.6.5: When selecting subsets of a larger set in which order is not import...
 11.6.6: The number of combinations of elements taken at a time is given by ...
 11.6.7: An odd integer
 11.6.8: An even integer
 11.6.9: A prime integer
 11.6.10: An integer that is greater than 9
 11.6.11: An integer that is divisible by 4
 11.6.12: An integer that is divisible by 3
 11.6.13: Two distinct integers whose sum is 9
 11.6.14: Two distinct integers whose sum is 8
 11.6.15: ENTERTAINMENT SYSTEMS A customer can choose one of three amplifiers...
 11.6.16: JOB APPLICANTS A college needs two additional faculty members: a ch...
 11.6.17: COURSE SCHEDULE A college student is preparing a course schedule fo...
 11.6.18: AIRCRAFT BOARDING Eight people are boarding an aircraft. Two have t...
 11.6.19: TRUEFALSE EXAM In how many ways can a sixquestion truefalse exam...
 11.6.20: TRUEFALSE EXAM In how many ways can a 12question truefalse exam ...
 11.6.21: LICENSE PLATE NUMBERS In the state of Pennsylvania, each standard a...
 11.6.22: LICENSE PLATE NUMBERS In a certain state, each automobile license p...
 11.6.23: LICENSE PLATE NUMBERS In a certain state, each automobile license p...
 11.6.24: FOURDIGIT NUMBERS How many fourdigit numbers can be formed under ...
 11.6.25: COMBINATION LOCK A combination lock will open when the right choice...
 11.6.26: COMBINATION LOCK A combination lock will open when the right choice...
 11.6.27: COMBINATION LOCK A combination lock willopen when the right choice ...
 11.6.28: COMBINATION LOCK A combination lock will open when the right choice...
 11.6.29: 4P4
 11.6.30: 5P5
 11.6.31: 8P3
 11.6.32: 20P2
 11.6.33: 5P4
 11.6.34: 7P4
 11.6.35: 20P5
 11.6.36: 100P5
 11.6.37: 100P3
 11.6.38: 10P8
 11.6.39: POSING FOR A PHOTOGRAPH In how many ways can five children posing f...
 11.6.40: RIDING IN A CAR In how many ways can six people sit in a sixpassen...
 11.6.41: CHOOSING OFFICERS From a pool of 12 candidates, the offices of pres...
 11.6.42: ASSEMBLY LINE PRODUCTION There are four processes involved in assem...
 11.6.43: A, A, G, E, E, E, M
 11.6.44: B, B, B, T, T, T, T, T
 11.6.45: A, L, G, E, B, R, A
 11.6.46: M, I, S, S, I, S, S, I, P, P, I
 11.6.47: Write all permutations of the letters A, B, C, and D.
 11.6.48: Write all permutations of the letters A, B, C, and D if the letters...
 11.6.49: BATTING ORDER A baseball coach is creating a nineplayer batting or...
 11.6.50: ATHLETICS Eight sprinters have qualified for the finals in the 100...
 11.6.51: 5C2
 11.6.52: 6C3
 11.6.53: 4C1
 11.6.54: 5C1
 11.6.55: 25C0
 11.6.56: 25C0
 11.6.57: 20C4
 11.6.58: 10C7
 11.6.59: 42C5
 11.6.60: 50C6
 11.6.61: Write all possible selections of two letters that can be formed fro...
 11.6.62: FORMING AN EXPERIMENTAL GROUP In order to conduct an experiment, fi...
 11.6.63: JURY SELECTION From a group of 40 people, a jury of 12 people is to...
 11.6.64: COMMITTEE MEMBERS A U.S. Senate Committee has 14 members. Assuming ...
 11.6.65: LOTTERY CHOICES In the Massachusetts Mass Cash game, a player choos...
 11.6.66: LOTTERY CHOICES In the Louisiana Lotto game, a player chooses six d...
 11.6.67: DEFECTIVE UNITS A shipment of 25 television sets contains three def...
 11.6.68: INTERPERSONAL RELATIONSHIPS The complexity of interpersonal relatio...
 11.6.69: POKER HAND You are dealt five cards from an ordinary deck of 52 pla...
 11.6.70: JOB APPLICANTS A clothing manufacturer interviews 12 people for fou...
 11.6.71: FORMING A COMMITTEE A sixmember research committee at a local coll...
 11.6.72: LAW ENFORCEMENT A police department uses computer imaging to create...
 11.6.73: Pentagon
 11.6.74: Hexagon
 11.6.75: Octagon
 11.6.76: Decagon (10 sides)
 11.6.77: GEOMETRY Three points that are not collinear determine three lines....
 11.6.78: LOTTERY Powerball is a lottery game that is operated by the MultiS...
 11.6.79: 14 nP3 n 2P4
 11.6.80: nP5 18 n2P4
 11.6.81: nP4 10 n1P3 n
 11.6.82: nP6 12 n1P5
 11.6.83: n 1P3 4 nP2
 11.6.84: n 2P3 6 n 2P1
 11.6.85: 4 n 1P2 n 2P3
 11.6.86: 5 n1P1 nP2
 11.6.87: The number of letter pairs that can be formed in any order from any...
 11.6.88: The number of permutations of elements can be determined by using t...
 11.6.89: What is the relationship between and ?
 11.6.90: Without calculating the numbers, determine which of the following i...
 11.6.91: nPn 1 nPn n
 11.6.92: nCn nC0
 11.6.93: nCn 1 nC1
 11.6.94: nCr nPr r!
 11.6.95: THINK ABOUT IT Can your calculator evaluate If not, explain why.
 11.6.96: CAPSTONE
 11.6.97: WRITING Explain in words the meaning of nPr .
Solutions for Chapter 11.6: Counting Principles
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 11.6: Counting Principles
Get Full SolutionsSince 97 problems in chapter 11.6: Counting Principles have been answered, more than 47289 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. Chapter 11.6: Counting Principles includes 97 full stepbystep solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. This expansive textbook survival guide covers the following chapters and their solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Particular solution x p.
Any solution to Ax = b; often x p has free variables = 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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

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).

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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}.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).