 10.2.1: 0! = ; .
 10.2.2: True or False n! = 1n + 12! n .
 10.2.3: A(n) is an ordered arrangement of r objects chosen from n objects.
 10.2.4: A(n) is an arrangement of r objects chosen from n distinct objects,...
 10.2.5: P1n, r2 =
 10.2.6: C1n, r2 =
 10.2.7: P16, 22
 10.2.8: P17, 22
 10.2.9: P14, 42
 10.2.10: P18, 82
 10.2.11: P17, 02
 10.2.12: P19, 02
 10.2.13: P18, 42
 10.2.14: P18, 32
 10.2.15: C18, 22
 10.2.16: C18, 62
 10.2.17: C17, 42
 10.2.18: C16, 22
 10.2.19: C115, 152
 10.2.20: C118, 12
 10.2.21: C126, 132
 10.2.22: C118, 92
 10.2.23: List all the ordered arrangements of 5 objects a, b, c, d, and e ch...
 10.2.24: List all the ordered arrangements of 5 objects a, b, c, d, and e ch...
 10.2.25: List all the ordered arrangements of 4 objects 1, 2, 3, and 4 choos...
 10.2.26: List all the ordered arrangements of 6 objects 1, 2, 3, 4, 5, and 6...
 10.2.27: List all the combinations of 5 objects a, b, c, d, and e taken 3 at...
 10.2.28: List all the combinations of 5 objects a, b, c, d, and e taken 2 at...
 10.2.29: List all the combinations of 4 objects 1, 2, 3, and 4 taken 3 at a ...
 10.2.30: List all the combinations of 6 objects 1, 2, 3, 4, 5, and 6 taken 3...
 10.2.31: Forming Codes How many twoletter codes can be formed using the let...
 10.2.32: Forming Codes How many twoletter codes can be formed using the let...
 10.2.33: Forming Numbers How many threedigit numbers can be formed using th...
 10.2.34: Forming Numbers How many threedigit numbers can be formed using th...
 10.2.35: Lining People Up In how many ways can 4 people be lined up?
 10.2.36: Stacking Boxes In how many ways can 5 different boxes be stacked?
 10.2.37: Forming Codes How many different threeletter codes are there if on...
 10.2.38: Forming Codes How many different fourletter codes are there if onl...
 10.2.39: Stocks on the NYSE Companies whose stocks are listed on the New Yor...
 10.2.40: Stocks on the NASDAQ Companies whose stocks are listed on the NASDA...
 10.2.41: Establishing Committees In how many ways can a committee of 4 stude...
 10.2.42: Establishing Committees In how many ways can a committee of 3 profe...
 10.2.43: Possible Answers on a True/False Test How many arrangements of answ...
 10.2.44: Possible Answers on a Multiplechoice Test How many arrangements of...
 10.2.45: Arranging Books Five different mathematics books are to be arranged...
 10.2.46: Forming License Plate Numbers How many different license plate numb...
 10.2.47: Birthday how many ways can 2 people each have different birthdays? ...
 10.2.48: Birthday how many ways can 5 people each have different birthdays? ...
 10.2.49: Forming a Committee A student dance committee is to be formed consi...
 10.2.50: Forming a Committee The student relations committee of a college co...
 10.2.51: Forming Words How many different 9letter words (real or imaginary)...
 10.2.52: Forming Words How many different 11letter words (real or imaginary...
 10.2.53: Selecting Objects An urn contains 7 white balls and 3 red balls. Th...
 10.2.54: Selecting Objects An urn contains 15 red balls and 10 white balls. ...
 10.2.55: Senate Committees The U.S. Senate has 100 members. Suppose that it ...
 10.2.56: Football Teams A defensive football squad consists of 25 players. O...
 10.2.57: Baseball In the American Baseball League, a designated hitter may b...
 10.2.58: Baseball In the National Baseball League, the pitcher usually bats ...
 10.2.59: Baseball Teams A baseball team has 15 members. Four of the players ...
 10.2.60: World Series In the World Series the American League team and the N...
 10.2.61: Basketball Teams A basketball team has 6 players who play guard (2 ...
 10.2.62: Basketball Teams On a basketball team of 12 players, 2 only play ce...
 10.2.63: Combination Locks A combination lock displays 50 numbers. To open i...
 10.2.64: Create a problem different from any found in the text that requires...
 10.2.65: Create a problem different from any found in the text that requires...
 10.2.66: Explain the difference between a permutation and a combination. Giv...
Solutions for Chapter 10.2: Permutations and Combinations
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 10.2: Permutations and Combinations
Get Full SolutionsCollege Algebra was written by and is associated to the ISBN: 9780321716811. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.2: Permutations and Combinations includes 66 full stepbystep solutions. Since 66 problems in chapter 10.2: Permutations and Combinations have been answered, more than 36596 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra, edition: 9.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.