 8.6.1: In Exercises 18, use the fommla for 11P, to evaluate each expressi...
 8.6.2: In Exercises 18, use the fommla for 11P, to evaluate each expressi...
 8.6.3: In Exercises 18, use the fommla for 11P, to evaluate each expressi...
 8.6.4: In Exercises 18, use the fommla for 11P, to evaluate each expressi...
 8.6.5: In Exercises 18, use the fommla for 11P, to evaluate each expressi...
 8.6.6: In Exercises 18, use the fommla for 11P, to evaluate each expressi...
 8.6.7: In Exercises 18, use the fommla for 11P, to evaluate each expressi...
 8.6.8: In Exercises 18, use the fommla for 11P, to evaluate each expressi...
 8.6.9: In Exercises 916. use the formula for ,.C, to evaluate each expres...
 8.6.10: In Exercises 916. use the formula for ,.C, to evaluate each expres...
 8.6.11: In Exercises 916. use the formula for ,.C, to evaluate each expres...
 8.6.12: In Exercises 916. use the formula for ,.C, to evaluate each expres...
 8.6.13: In Exercises 916. use the formula for ,.C, to evaluate each expres...
 8.6.14: In Exercises 916. use the formula for ,.C, to evaluate each expres...
 8.6.15: In Exercises 916. use the formula for ,.C, to evaluate each expres...
 8.6.16: In Exercises 916. use the formula for ,.C, to evaluate each expres...
 8.6.17: In Exercises 1720, does the problem hwolve permutations or combina...
 8.6.18: In Exercises 1720, does the problem hwolve permutations or combina...
 8.6.19: In Exercises 1720, does the problem hwolve permutations or combina...
 8.6.20: In Exercises 1720, does the problem hwolve permutations or combina...
 8.6.21: In Exercises 2128, evaluate each expression. p 3 2Dp2;;1  7C,
 8.6.22: In Exercises 2128, evaluate each expression. 2Dp2 ;;1  ,   wC...
 8.6.23: In Exercises 2128, evaluate each expression. 1  ,P, ,P
 8.6.24: In Exercises 2128, evaluate each expression. 1  5P3 10P4
 8.6.25: In Exercises 2128, evaluate each expression. ,c,  2!!!. ,c, 96!
 8.6.26: In Exercises 2128, evaluate each expression. 10C3  46! 44!
 8.6.27: In Exercises 2128, evaluate each expression. 4C2 . 6C1 18C
 8.6.28: In Exercises 2128, evaluate each expression. 5C1.7C2 C3
 8.6.29: Use the Fundamental Counting Principle to solve Exercises 2940. Th...
 8.6.30: Use the Fundamental Counting Principle to solve Exercises 2940. A ...
 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 ...
 8.6.33: Use the Fundamental Counting Principle to solve Exercises 2940. Yo...
 8.6.34: Use the Fundamental Counting Principle to solve Exercises 2940. Yo...
 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. Ho...
 8.6.37: Use the Fundamental Counting Principle to solve Exercises 2940. Si...
 8.6.38: Use the Fundamental Counting Principle to solve Exercises 2940. Fi...
 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 ...
 8.6.41: Use the formula for "Pr to solve Exercises 4148. A club with ten m...
 8.6.42: Use the formula for "Pr to solve Exercises 4148. A corporation has...
 8.6.43: Use the formula for "Pr to solve Exercises 4148. For a segment of ...
 8.6.44: Use the formula for "Pr to solve Exercises 4148. Suppose you are a...
 8.6.45: Use the formula for "Pr to solve Exercises 4148. In a race in whic...
 8.6.46: Use the formula for "Pr to solve Exercises 4148. In a production o...
 8.6.47: Use the formula for "Pr to solve Exercises 4148. Nine bands have v...
 8.6.48: Use the formula for "Pr to solve Exercises 4148. How many arrangem...
 8.6.49: Use the formula for "Cr to solve Exercises 4956. An e lection ba l...
 8.6.50: Use the formula for "Cr to solve Exercises 4956. A four.pe rson co...
 8.6.51: Use the formula for "Cr to solve Exercises 4956. Of I 2 possible b...
 8.6.52: Use the formula for "Cr to solve Exercises 4956. 111ere a re 14 st...
 8.6.53: Use the formula for "Cr to solve Exercises 4956. You volunteer to ...
 8.6.54: Use the formula for "Cr to solve Exercises 4956. Of the 100 people...
 8.6.55: Use the formula for "Cr to solve Exercises 4956. To win al LOTIO i...
 8.6.56: Use the formula for "Cr to solve Exercises 4956. To \\~n in the Ne...
 8.6.57: In Exercises 5766, solve by the method of your choice. In a race i...
 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 r...
 8.6.60: In Exercises 5766, solve by the method of your choice. Fifty peopl...
 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 peopl...
 8.6.63: In Exercises 5766, solve by the method of your choice. How many di...
 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 fl...
 8.6.66: In Exercises 5766, solve by the method of your choice. BaskinRobb...
 8.6.67: Exercises 6772 are based on the following jokes abouJ books: "Outs...
 8.6.68: Exercises 6772 are based on the following jokes abouJ books: "Outs...
 8.6.69: Exercises 6772 are based on the following jokes abouJ books: "Outs...
 8.6.70: Exercises 6772 are based on the following jokes abouJ books: "Outs...
 8.6.71: Exercises 6772 are based on the following jokes abouJ books: "Outs...
 8.6.72: Exercises 6772 are based on the following jokes abouJ books: "Outs...
 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 "P, represents.
 8.6.77: Write a word problem tha t 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 !ha t can be solved by evaluating 7C3
 8.6.81: Use a graphing utility with a n ~ me nu item to verify your answers...
 8.6.82: Use a graphing utility with an l:fJ me nu item to verify your answe...
 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 tme or fals...
 8.6.88: In Exercises 8790, determine whether each statement is tme or fals...
 8.6.89: In Exercises 8790, determine whether each statement is tme or fals...
 8.6.90: In Exercises 8790, determine whether each statement is tme 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 fomed using t...
 8.6.93: A mathematics exam consists of 10 multiplechoice questions and 5 o...
 8.6.94: The group should select real.world situation.'i where the Fw1dament...
 8.6.95: Exercises 9597 will help you prepare for the material covered in t...
 8.6.96: Exercises 9597 will help you prepare for the material covered in t...
 8.6.97: Exercises 9597 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  6th Edition
ISBN: 9780321782281
Solutions for Chapter 8.6: Counting Principles, Permutations, and Combinations
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra , edition: 6. Since 97 problems in chapter 8.6: Counting Principles, Permutations, and Combinations have been answered, more than 35464 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321782281. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.6: Counting Principles, Permutations, and Combinations includes 97 full stepbystep solutions.

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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).

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.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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.

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.