 8.6.26: COMBINATION LOCK A combination lock willopen when the right choice ...
 8.6.27: CONCERT SEATS Four couples have reserved seats ina row for a concer...
 8.6.28: SINGLE FILE In how many orders can four girls andfour boys walk thr...
 8.6.29: In Exercises 2934, evaluate nPr .4P4
 8.6.30: In Exercises 2934, evaluate nPr .5P5
 8.6.31: In Exercises 2934, evaluate nPr .8P3
 8.6.32: In Exercises 2934, evaluate nPr .0P2
 8.6.33: In Exercises 2934, evaluate nPr . 5P4
 8.6.34: In Exercises 2934, evaluate nPr .7P4
 8.6.35: In Exercises 3538, evaluate using a graphing utility. 20P5
 8.6.36: In Exercises 3538, evaluate using a graphing utility. 100P5
 8.6.37: In Exercises 3538, evaluate using a graphing utility. 100P3
 8.6.38: In Exercises 3538, evaluate using a graphing utility. 10P8
 8.6.39: POSING FOR A PHOTOGRAPH In how many wayscan five children posing fo...
 8.6.40: RIDING IN A CAR In how many ways can six peoplesit in a sixpasseng...
 8.6.41: CHOOSING OFFICERS From a pool of 12 candidates,the offices of presi...
 8.6.42: ASSEMBLY LINE PRODUCTION There are fourprocesses involved in assemb...
 8.6.43: In Exercises 4346, find the number of distinguishablepermutations o...
 8.6.44: In Exercises 4346, find the number of distinguishablepermutations o...
 8.6.45: In Exercises 4346, find the number of distinguishablepermutations o...
 8.6.46: In Exercises 4346, find the number of distinguishablepermutations o...
 8.6.47: Write all permutations of the letters A, B, C, and D.
 8.6.48: Write all permutations of the letters A, B, C, and D ifthe letters ...
 8.6.49: BATTING ORDER A baseball coach is creating anineplayer batting ord...
 8.6.50: ATHLETICS Eight sprinters have qualified for thefinals in the 100m...
 8.6.51: In Exercises 5156, evaluate using the formula from thissection. 5C2
 8.6.52: In Exercises 5156, evaluate using the formula from thissection. 6C3
 8.6.53: In Exercises 5156, evaluate using the formula from thissection. 4C1
 8.6.54: In Exercises 5156, evaluate using the formula from thissection. 5C1
 8.6.55: In Exercises 5156, evaluate using the formula from thissection. 25C0
 8.6.56: In Exercises 5156, evaluate using the formula from thissection. 20C0
 8.6.57: In Exercises 5760, evaluate using a graphing utility.20C4
 8.6.58: In Exercises 5760, evaluate using a graphing utility. 10C7
 8.6.59: In Exercises 5760, evaluate using a graphing utility. 42C5
 8.6.60: In Exercises 5760, evaluate using a graphing utility. 50C6
 8.6.61: Write all possible selections of two letters that can beformed from...
 8.6.62: FORMING AN EXPERIMENTAL GROUP In order toconduct an experiment, fiv...
 8.6.63: JURY SELECTION From a group of 40 people, a juryof 12 people is to ...
 8.6.64: COMMITTEE MEMBERS A U.S. Senate Committeehas 14 members. Assuming p...
 8.6.65: LOTTERY CHOICES In the Massachusetts Mass Cashgame, a player choose...
 8.6.66: LOTTERY CHOICES In the Louisiana Lotto game, aplayer chooses six di...
 8.6.67: DEFECTIVE UNITS A shipment of 25 television setscontains three defe...
 8.6.68: INTERPERSONAL RELATIONSHIPS The complexityof interpersonal relation...
 8.6.69: POKER HAND You are dealt five cards from anordinary deck of 52 play...
 8.6.70: JOB APPLICANTS A clothing manufacturer interviews12 people for four...
 8.6.71: FORMING A COMMITTEE A sixmember researchcommittee at a local colle...
 8.6.72: LAW ENFORCEMENT A police department usescomputer imaging to create ...
 8.6.73: In Exercises 7376, find the number ofdiagonals of the polygon. (A l...
 8.6.74: In Exercises 7376, find the number ofdiagonals of the polygon. (A l...
 8.6.75: In Exercises 7376, find the number ofdiagonals of the polygon. (A l...
 8.6.76: In Exercises 7376, find the number ofdiagonals of the polygon. (A l...
 8.6.77: GEOMETRY Three points that are not collineardetermine three lines. ...
 8.6.78: LOTTERY Powerball is a lottery game that is operatedby the MultiSt...
 8.6.79: In Exercises 7986, solve for n 14 nP3 n2P4
 8.6.80: In Exercises 7986, solve for n nP5 18 n2P4
 8.6.81: In Exercises 7986, solve for n P4 10 n1P3
 8.6.82: In Exercises 7986, solve for nnn P6 12 n1P5
 8.6.83: In Exercises 7986, solve for n n1P3 4 n n2P3 6 n2P1 P2
 8.6.84: In Exercises 7986, solve for n n2P3 6 n2P1
 8.6.85: In Exercises 7986, solve for n 4 n1P2 n2P3
 8.6.86: In Exercises 7986, solve for n 5 n1P1 nP2
 8.6.87: TRUE OR FALSE? In Exercises 87 and 88, determinewhether the stateme...
 8.6.88: TRUE OR FALSE? In Exercises 87 and 88, determinewhether the stateme...
 8.6.89: What is the relationship between and ?
 8.6.90: Without calculating the numbers, determine which ofthe following is...
 8.6.91: PROOF In Exercises 9194, prove the identity.
 8.6.92: PROOF In Exercises 9194, prove the identity.
 8.6.93: PROOF In Exercises 9194, prove the identity.
 8.6.94: PROOF In Exercises 9194, prove the identity.
 8.6.95: THINK ABOUT IT Can your calculator evaluateIf not, explain why.
 8.6.96: CAPSTONE Decide whether each scenario shouldbe counted using permut...
 8.6.97: WRITING Explain in words the meaning of nPr .
Solutions for Chapter 8.6: Counting Principles
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 8.6: Counting Principles
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9781439048696. Since 72 problems in chapter 8.6: Counting Principles have been answered, more than 29457 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra , edition: 8. Chapter 8.6: Counting Principles includes 72 full stepbystep solutions.

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

Column space C (A) =
space of all combinations of the columns of A.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

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.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Outer product uv T
= column times row = rank one matrix.

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.

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.