 5.3.5.3.1: List all the permutations of {a , b, e}.
 5.3.5.3.2: How many different permutations are there of the set {a, b, e, d, e...
 5.3.5.3.3: How many permutations of {a , b, e, d, e, t, g} end with a?
 5.3.5.3.4: Let S = {I, 2, 3, 4, 5}. a) List all the 3permutations of S. b) Li...
 5.3.5.3.5: Find the value of each of these quantities. a) P(6, 3) b) P(6, 5) c...
 5.3.5.3.6: Find the value of each of these quantities. a) C(5, 1) b) C(5, 3) c...
 5.3.5.3.7: Find the number of 5permutations of a set with nine elements.
 5.3.5.3.8: In how many different orders can five runners finish a race if no t...
 5.3.5.3.9: How many possibilities are there for the win, place, and show (firs...
 5.3.5.3.10: There are six different candidates for governor of a state. In how ...
 5.3.5.3.11: How many bit strings oflength 10 contain a) exactly four Is? b) at ...
 5.3.5.3.12: How many bit strings oflength 12 contain a) exactly three Is? b) at...
 5.3.5.3.13: A group contains n men and n women. How many ways are there to arra...
 5.3.5.3.14: In how many ways can a set of two positive integers less than 100 b...
 5.3.5.3.15: In how many ways can a set of five letters be selected from the Eng...
 5.3.5.3.16: How many subsets with an odd number of elements does a set with 10 ...
 5.3.5.3.17: How many subsets with more than two elements does a set with 100 el...
 5.3.5.3.18: A coin is flipped eight times where each flip comes up either heads...
 5.3.5.3.19: A coin is flipped 10 times where each flip comes up either heads or...
 5.3.5.3.20: How many bit strings oflength 10 have a) exactly three Os? b) more ...
 5.3.5.3.21: How many permutations of the letters ABCDEFG contain a) the string ...
 5.3.5.3.22: How many permutations of the letters ABCDEFGH contain a) the string...
 5.3.5.3.23: How many ways are there for eight men and five women to stand in a ...
 5.3.5.3.24: How many ways are there for 10 women and six men to stand in a line...
 5.3.5.3.25: One hundred tickets, numbered I, 2, 3, ... , 100, are sold to 100 d...
 5.3.5.3.26: Thirteen people on a softball team show up for a game. a) How many ...
 5.3.5.3.27: A club has 25 members. a) How many ways are there to choose four me...
 5.3.5.3.28: A professor writes 40 discrete mathematics true/false questions. Of...
 5.3.5.3.29: How many 4permutations of the positive integers not exceeding 100 ...
 5.3.5.3.30: Seven women and nine men are on the faculty in the mathematics depa...
 5.3.5.3.31: The English alphabet contains 21 consonants and five vowels. How ma...
 5.3.5.3.32: How many strings of six lowercase letters from the English alphabet...
 5.3.5.3.33: Suppose that a department contains 10 men and 15 women. How many wa...
 5.3.5.3.34: Suppose that a department contains 10 men and 15 women. How many wa...
 5.3.5.3.35: How many bit strings contain exactly eight Os and lO is if every 0 ...
 5.3.5.3.36: How many bit strings contain exactly five Os and 14 Is if every 0 m...
 5.3.5.3.37: How many bit strings of length 10 contain at least three Is and at ...
 5.3.5.3.38: How many ways are there to select 12 countries in the United Nation...
 5.3.5.3.39: How many license plates consisting of three letters followed by thr...
 5.3.5.3.40: How many ways are there to seat six people around a circular table,...
 5.3.5.3.41: How many ways are there for a horse race with three horses to finis...
 5.3.5.3.42: How many ways are there for a horse race with four horses to finish...
 5.3.5.3.43: There are six runners in the 100yard dash. How many ways are there...
 5.3.5.3.44: This procedure is used to break ties in games in the championship r...
Solutions for Chapter 5.3: Counting
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 5.3: Counting
Get Full SolutionsSince 44 problems in chapter 5.3: Counting have been answered, more than 35418 students have viewed full stepbystep solutions from this chapter. Chapter 5.3: Counting includes 44 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720.

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!

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

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

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

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

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

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).