 Chapter 11.1: Fill in each blank so that the resulting statement is true.If you c...
 Chapter 11.2: Fill in each blank so that the resulting statement is true.The numb...
 Chapter 11.3: In 34, determine whether each statement is true or false. If the st...
 Chapter 11.4: In 34, determine whether each statement is true or false. If the st...
 Chapter 11.5: In 47, determine whether each statement is true or false. If the st...
 Chapter 11.6: In 47, determine whether each statement is true or false. If the st...
 Chapter 11.7: In 47, determine whether each statement is true or false. If the st...
 Chapter 11.8: In 58, determine whether each statement is true or false. If the st...
 Chapter 11.9: In 69, determine whether each statement is true or false. If the st...
 Chapter 11.10: In 710, evaluate each factorial expression. 11!(11  3P)!
 Chapter 11.11: In 1112, use the formula for nPr to evaluate each expression 10P6
 Chapter 11.12: In 1112, use the formula for nPr to evaluate each expression 100P2
 Chapter 11.13: In 1314, use the formula for nCr to evaluate each expression. 11C7
 Chapter 11.14: In 1314, use the formula for nCr to evaluate each expression. 14C5
 Chapter 11.15: In 1517, does the problem involve permutations or combinations? Exp...
 Chapter 11.16: In 1517, does the problem involve permutations or combinations? Exp...
 Chapter 11.17: In 1517, does the problem involve permutations or combinations? Exp...
 Chapter 11.18: In 1828, solve each problem using an appropriate method.Six acts ar...
 Chapter 11.19: In 1828, solve each problem using an appropriate method. A club wit...
 Chapter 11.20: In 1828, solve each problem using an appropriate method. An electio...
 Chapter 11.21: In 1828, solve each problem using an appropriate method.In how many...
 Chapter 11.22: In 1828, solve each problem using an appropriate method. From the 2...
 Chapter 11.23: In 1828, solve each problem using an appropriate method.You need to...
 Chapter 11.24: In 1828, solve each problem using an appropriate method.Suppose you...
 Chapter 11.25: In 1828, solve each problem using an appropriate method.In how many...
 Chapter 11.26: In 1828, solve each problem using an appropriate method.How many di...
 Chapter 11.27: In 1828, solve each problem using an appropriate method.A political...
 Chapter 11.28: In 1828, solve each problem using an appropriate method.In how many...
 Chapter 11.29: In 2932, a die is rolled. Find the probability of rolling a 6.
 Chapter 11.30: In 2932, a die is rolled. Find the probability of rolling a number ...
 Chapter 11.31: In 2932, a die is rolled. Find the probability of rolling a number ...
 Chapter 11.32: In 2932, a die is rolled. Find the probability of rolling a number ...
 Chapter 11.33: In 3337, you are dealt one card from a 52card deck. Find the proba...
 Chapter 11.34: In 3337, you are dealt one card from a 52card deck. Find the proba...
 Chapter 11.35: In 3337, you are dealt one card from a 52card deck. Find the proba...
 Chapter 11.36: In 3337, you are dealt one card from a 52card deck. Find the proba...
 Chapter 11.37: In 3337, you are dealt one card from a 52card deck. Find the proba...
 Chapter 11.38: In 3840, suppose that you reach into a bag and randomly select one ...
 Chapter 11.39: In 3840, suppose that you reach into a bag and randomly select one ...
 Chapter 11.40: In 3840, suppose that you reach into a bag and randomly select one ...
 Chapter 11.41: TaySachs disease occurs in 1 of every 3600 births among Jews from ...
 Chapter 11.42: The table shows the employment status of the U.S. civilian labor fo...
 Chapter 11.43: The table shows the employment status of the U.S. civilian labor fo...
 Chapter 11.44: The table shows the employment status of the U.S. civilian labor fo...
 Chapter 11.45: If cities A, B, C, and D are visited in random order, each city vis...
 Chapter 11.46: In 4649, suppose that six singers are being lined up to perform at ...
 Chapter 11.47: In 4649, suppose that six singers are being lined up to perform at ...
 Chapter 11.48: In 4649, suppose that six singers are being lined up to perform at ...
 Chapter 11.49: In 4649, suppose that six singers are being lined up to perform at ...
 Chapter 11.50: A lottery game is set up so that each player chooses five different...
 Chapter 11.51: A committee of four people is to be selected from six Democrats and...
 Chapter 11.52: If you are dealt 3 cards from a shuffled deck of red cards (26 diff...
 Chapter 11.53: In 5357, a die is rolled. Find the probability of not rolling a 5.
 Chapter 11.54: In 5357, a die is rolled. Find the probability of not rolling a num...
 Chapter 11.55: In 5357, a die is rolled. Find the probability of rolling a 3 or a 5.
 Chapter 11.56: In 5357, a die is rolled. Find the probability of rolling a number ...
 Chapter 11.57: In 5357, a die is rolled. Find the probability of rolling a number ...
 Chapter 11.58: In 5863, you draw one card from a 52card deck. Find the probabilit...
 Chapter 11.59: In 5863, you draw one card from a 52card deck. Find the probabilit...
 Chapter 11.60: In 5863, you draw one card from a 52card deck. Find the probabilit...
 Chapter 11.61: In 5863, you draw one card from a 52card deck. Find the probabilit...
 Chapter 11.62: In 5863, you draw one card from a 52card deck. Find the probabilit...
 Chapter 11.63: In 5863, you draw one card from a 52card deck. Find the probabilit...
 Chapter 11.64: In 6469, it is equally probable that the pointer on the spinner sho...
 Chapter 11.65: In 6469, it is equally probable that the pointer on the spinner sho...
 Chapter 11.66: In 6469, it is equally probable that the pointer on the spinner sho...
 Chapter 11.67: In 6469, it is equally probable that the pointer on the spinner sho...
 Chapter 11.68: In 6469, it is equally probable that the pointer on the spinner sho...
 Chapter 11.69: In 6469, it is equally probable that the pointer on the spinner sho...
 Chapter 11.70: Use this information to solve 7071. At a workshop on police work an...
 Chapter 11.71: Use this information to solve 7071. At a workshop on police work an...
 Chapter 11.72: Suppose that a survey of 350 college students is taken. Each studen...
 Chapter 11.73: Suppose that a survey of 350 college students is taken. Each studen...
 Chapter 11.74: Suppose that a survey of 350 college students is taken. Each studen...
 Chapter 11.75: Suppose that a survey of 350 college students is taken. Each studen...
 Chapter 11.76: One card is randomly selected from a deck of 52 cards. Find the odd...
 Chapter 11.77: The winner of a raffle will receive a twoyear scholarship to any c...
 Chapter 11.78: The odds in favor of a candidate winning an election are given at 3...
 Chapter 11.79: Use the spinner shown to solve 7983. It is equally likely that the ...
 Chapter 11.80: Use the spinner shown to solve 7983. It is equally likely that the ...
 Chapter 11.81: Use the spinner shown to solve 7983. It is equally likely that the ...
 Chapter 11.82: If the pointer is spun three times, find the probability it will la...
 Chapter 11.83: If the pointer is spun three times, find the probability it will la...
 Chapter 11.84: What is the probability of a family having five boys born ina row?
 Chapter 11.85: The probability of a flood in any given year in a region prone to f...
 Chapter 11.86: In 8687, two students are selected from a group of four psychology ...
 Chapter 11.87: In 8687, two students are selected from a group of four psychology ...
 Chapter 11.88: A final visit to the box of chocolates: Its now grown to a box of 5...
 Chapter 11.89: A single die is tossed. Find the probability that the tossed die sh...
 Chapter 11.90: A letter is randomly selected from the letters of the English alpha...
 Chapter 11.91: The following numbers are each written on a colored chip. The chips...
 Chapter 11.92: Find the probability that a randomly selected person from this grou...
 Chapter 11.93: Find the probability that a randomly selected person from this grou...
 Chapter 11.94: Find the probability that a randomly selected person from this grou...
 Chapter 11.95: Find the probability that a randomly selected person from this grou...
 Chapter 11.96: Find the probability that a randomly selected person from this grou...
 Chapter 11.97: Find the probability that a randomly selected person from this grou...
 Chapter 11.98: Suppose that two people are randomly selected, in succession, from ...
 Chapter 11.99: Suppose that two people are randomly selected, in succession, from ...
 Chapter 11.100: In 100106, use the data in the table to find the probability, expre...
 Chapter 11.101: In 100106, use the data in the table to find the probability, expre...
 Chapter 11.102: In 100106, use the data in the table to find the probability, expre...
 Chapter 11.103: In 100106, use the data in the table to find the probability, expre...
 Chapter 11.104: In 100106, use the data in the table to find the probability, expre...
 Chapter 11.105: In 100106, use the data in the table to find the probability, expre...
 Chapter 11.106: In 100106, use the data in the table to find the probability, expre...
 Chapter 11.107: The numbers that the pointer can land on and their respective proba...
Solutions for Chapter Chapter 11: Counting Methods and Probability Theory
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter Chapter 11: Counting Methods and Probability Theory
Get Full SolutionsChapter Chapter 11: Counting Methods and Probability Theory includes 107 full stepbystep solutions. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Since 107 problems in chapter Chapter 11: Counting Methods and Probability Theory have been answered, more than 67243 students have viewed full stepbystep solutions from this chapter.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

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.

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 inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

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

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.