 8.6.1: Fill in each blank so that the resulting statement is true. If you ...
 8.6.2: Fill in each blank so that the resulting statement is true. The num...
 8.6.3: Fill in each blank so that the resulting statement is true. The num...
 8.6.4: Fill in each blank so that the resulting statement is true. The num...
 8.6.5: Fill in each blank so that the resulting statement is true. The for...
 8.6.6: In Exercises 18, use the formula for nPr to evaluate each expressio...
 8.6.7: In Exercises 18, use the formula for nPr to evaluate each expressio...
 8.6.8: In Exercises 18, use the formula for nPr to evaluate each expressio...
 8.6.9: In Exercises 916, use the formula for nCr to evaluate each expressi...
 8.6.10: In Exercises 916, use the formula for nCr to evaluate each expressi...
 8.6.11: In Exercises 916, use the formula for nCr to evaluate each expressi...
 8.6.12: In Exercises 916, use the formula for nCr to evaluate each expressi...
 8.6.13: In Exercises 916, use the formula for nCr to evaluate each expressi...
 8.6.14: In Exercises 916, use the formula for nCr to evaluate each expressi...
 8.6.15: In Exercises 916, use the formula for nCr to evaluate each expressi...
 8.6.16: In Exercises 916, use the formula for nCr to evaluate each expressi...
 8.6.17: In Exercises 1720, does the problem involve permutations or combina...
 8.6.18: In Exercises 1720, does the problem involve permutations or combina...
 8.6.19: In Exercises 1720, does the problem involve permutations or combina...
 8.6.20: In Exercises 1720, does the problem involve permutations or combina...
 8.6.21: In Exercises 2128, evaluate each expression. 7P33!  7C3
 8.6.22: In Exercises 2128, evaluate each expression. 20P22!  20C2
 8.6.23: In Exercises 2128, evaluate each expression. 1  3P24P3
 8.6.24: In Exercises 2128, evaluate each expression. 1  5P310P4
 8.6.25: In Exercises 2128, evaluate each expression. 7C35C4 98!96!
 8.6.26: In Exercises 2128, evaluate each expression. 10C36C4 46!44!
 8.6.27: In Exercises 2128, evaluate each expression. 4C2 #6C118C3
 8.6.28: In Exercises 2128, evaluate each expression. 5C1 #7C212C3
 8.6.29: Use the Fundamental Counting Principle to solve Exercises 2940. The...
 8.6.30: Use the Fundamental Counting Principle to solve Exercises 2940. A p...
 8.6.31: Use the Fundamental Counting Principle to solve Exercises 2940. An ...
 8.6.32: Use the Fundamental Counting Principle to solve Exercises 2940. A r...
 8.6.33: Use the Fundamental Counting Principle to solve Exercises 2940. You...
 8.6.34: Use the Fundamental Counting Principle to solve Exercises 2940. You...
 8.6.35: Use the Fundamental Counting Principle to solve Exercises 2940. In ...
 8.6.36: Use the Fundamental Counting Principle to solve Exercises 2940. How...
 8.6.37: Use the Fundamental Counting Principle to solve Exercises 2940. Six...
 8.6.38: Use the Fundamental Counting Principle to solve Exercises 2940. Fiv...
 8.6.39: Use the Fundamental Counting Principle to solve Exercises 2940. In ...
 8.6.40: Use the Fundamental Counting Principle to solve Exercises 2940. A t...
 8.6.41: Use the formula for nPr to solve Exercises 4148. A club with ten me...
 8.6.42: Use the formula for nPr to solve Exercises 4148. A corporation has ...
 8.6.43: Use the formula for nPr to solve Exercises 4148. For a segment of a...
 8.6.44: Use the formula for nPr to solve Exercises 4148. Suppose you are as...
 8.6.45: Use the formula for nPr to solve Exercises 4148. In a race in which...
 8.6.46: Use the formula for nPr to solve Exercises 4148. In a production of...
 8.6.47: Use the formula for nPr to solve Exercises 4148. Nine bands have vo...
 8.6.48: Use the formula for nPr to solve Exercises 4148. How many arrangeme...
 8.6.49: Use the formula for nCr to solve Exercises 4956. An election ballot...
 8.6.50: Use the formula for nCr to solve Exercises 4956. A fourperson comm...
 8.6.51: Use the formula for nCr to solve Exercises 4956. Of 12 possible boo...
 8.6.52: Use the formula for nCr to solve Exercises 4956. There are 14 stand...
 8.6.53: Use the formula for nCr to solve Exercises 4956. You volunteer to h...
 8.6.54: Use the formula for nCr to solve Exercises 4956. Of the 100 people ...
 8.6.55: Use the formula for nCr to solve Exercises 4956. To win at LOTTO in...
 8.6.56: Use the formula for nCr to solve Exercises 4956. To win in the New ...
 8.6.57: In Exercises 5766, solve by the method of your choice. In a race in...
 8.6.58: In Exercises 5766, solve by the method of your choice. A book club ...
 8.6.59: In Exercises 5766, solve by the method of your choice. A medical re...
 8.6.60: In Exercises 5766, solve by the method of your choice. Fifty people...
 8.6.61: In Exercises 5766, solve by the method of your choice. From a club ...
 8.6.62: In Exercises 5766, solve by the method of your choice. Fifty people...
 8.6.63: In Exercises 5766, solve by the method of your choice. How many dif...
 8.6.64: In Exercises 5766, solve by the method of your choice. Nine comedy ...
 8.6.65: In Exercises 5766, solve by the method of your choice. Using 15 fla...
 8.6.66: In Exercises 5766, solve by the method of your choice. BaskinRobbi...
 8.6.67: Exercises 6772 are based on the following jokes about books: Outsid...
 8.6.68: Exercises 6772 are based on the following jokes about books: Outsid...
 8.6.69: Exercises 6772 are based on the following jokes about books: Outsid...
 8.6.70: Exercises 6772 are based on the following jokes about books: Outsid...
 8.6.71: Exercises 6772 are based on the following jokes about books: Outsid...
 8.6.72: Exercises 6772 are based on the following jokes about books: Outsid...
 8.6.73: Explain the Fundamental Counting Principle.
 8.6.74: Write an original problem that can be solved using the Fundamental ...
 8.6.75: What is a permutation?
 8.6.76: Describe what nPr represents.
 8.6.77: Write a word problem that can be solved by evaluating 7P3.
 8.6.78: What is a combination?
 8.6.79: Explain how to distinguish between permutation and combination prob...
 8.6.80: Write a word problem that can be solved by evaluating 7C3.
 8.6.81: Use a graphing utility with an nPr menu item to verify your answers...
 8.6.82: Use a graphing utility with an nCr menu item to verify your answers...
 8.6.83: In Exercises 8386, determine whether each statement makes sense or ...
 8.6.84: In Exercises 8386, determine whether each statement makes sense or ...
 8.6.85: In Exercises 8386, determine whether each statement makes sense or ...
 8.6.86: In Exercises 8386, determine whether each statement makes sense or ...
 8.6.87: In Exercises 8790, determine whether each statement is true or fals...
 8.6.88: In Exercises 8790, determine whether each statement is true or fals...
 8.6.89: In Exercises 8790, determine whether each statement is true or fals...
 8.6.90: In Exercises 8790, determine whether each statement is true or fals...
 8.6.91: Five men and five women line up at a checkout counter in a store. I...
 8.6.92: How many fourdigit odd numbers less than 6000 can be formed using ...
 8.6.93: A mathematics exam consists of 10 multiplechoice questions and 5 o...
 8.6.94: The group should select realworld situations where the Fundamental...
 8.6.95: Solve and determine whether 8(x  3) + 4 = 8x  21 is an identity, ...
 8.6.96: If f(x) = 4x2  5x  2, find f(x + h)  f(x) h , h 0 and simplify. ...
 8.6.97: Expand: log7 a 1 5 x 49y10 b. (Section 4.3, Example 4)
 8.6.98: Exercises 98100 will help you prepare for the material covered in t...
 8.6.99: Exercises 98100 will help you prepare for the material covered in t...
 8.6.100: Exercises 98100 will help you prepare for the material covered in t...
Solutions for Chapter 8.6: Counting Principles, Permutations, and Combinations
Full solutions for College Algebra  7th Edition
ISBN: 9780134469164
Solutions for Chapter 8.6: Counting Principles, Permutations, and Combinations
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra , edition: 7. Chapter 8.6: Counting Principles, Permutations, and Combinations includes 100 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 100 problems in chapter 8.6: Counting Principles, Permutations, and Combinations have been answered, more than 32562 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780134469164.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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!).

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.

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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