 5.2.5.2.1: Show that in any set of six classes, each meeting regularly once a ...
 5.2.5.2.2: Show that if there are 30 students in a class, then at least two ha...
 5.2.5.2.3: A drawer contains a dozen brown socks and a dozen black socks, all ...
 5.2.5.2.4: A bowl contains 10 red balls and 10 blue balls. A woman selects bal...
 5.2.5.2.5: Show that among any group of five (not necessarily consecutive) int...
 5.2.5.2.6: Let d be a positive integer. Show that among any group of d + 1 (no...
 5.2.5.2.7: Let n be a positive integer. Show that in any set of n consecutive ...
 5.2.5.2.8: Show that if f is a function from S to T, where S and T are finite ...
 5.2.5.2.9: What is the minimum number of students, each of whom comes from one...
 5.2.5.2.10: Let (xi , Yi), i = 1 , 2, 3, 4, 5, be a set offive distinct points ...
 5.2.5.2.11: Let (xi , Yi,zi), i = 1 , 2, 3, 4, 5, 6, 7, 8, 9, be a set ofnine d...
 5.2.5.2.12: How many ordered pairs of integers (a, b) are needed to guarantee t...
 5.2.5.2.13: a) Show that if five integers are selected from the first eight pos...
 5.2.5.2.14: a) Show that if seven integers are selected from the first 10 posit...
 5.2.5.2.15: How many numbers must be selected from the set {I, 2, 3, 4, 5, 6} t...
 5.2.5.2.16: How many numbers must be selected from the set {I, 3, 5, 7, 9, 1 1 ...
 5.2.5.2.17: A company stores products in a warehouse. Storage bins in this ware...
 5.2.5.2.18: Suppose that there are nine students in a discrete mathematics clas...
 5.2.5.2.19: Suppose that every student in a discrete mathematics class of 25 st...
 5.2.5.2.20: Find an increasing subsequence of maximal length and a decreasing s...
 5.2.5.2.21: Construct a sequence of 16 positive integers that has no increasing...
 5.2.5.2.22: Show that if there are 10 I people of different heights standing in...
 5.2.5.2.23: Describe an algorithm in pseudocode for producing the largest incre...
 5.2.5.2.24: Show that in a group of five people (where any two people are eithe...
 5.2.5.2.25: Show that in a group of 10 people (where any two people are either ...
 5.2.5.2.26: Use Exercise 25 to show that among any group of 20 people (where an...
 5.2.5.2.27: Show that if n is a positive integer with n ::: 2, then the Ramsey ...
 5.2.5.2.28: Show that if m and n are positive integers with m ::: 2 and n ::: 2...
 5.2.5.2.29: Show that there are at least six people in California (population: ...
 5.2.5.2.30: Show that if there are 100,000,000 wage earners in the United State...
 5.2.5.2.31: There are 38 different time periods during which classes at a unive...
 5.2.5.2.32: A computer network consists of six computers. Each computer is dire...
 5.2.5.2.33: A computer network consists of six computers. Each computer is dire...
 5.2.5.2.34: Find the least number of cables required to connect eight computers...
 5.2.5.2.35: Find the least number of cables required to connect 100 computers t...
 5.2.5.2.36: Prove that at a party where there are at least two people, there ar...
 5.2.5.2.37: An arm wrestler is the champion for a period of75 hours. (Here, by ...
 5.2.5.2.38: Is the statement in Exercise 37 true if 24 is replaced by a) 2? b) ...
 5.2.5.2.39: Show that if f is a function from S to T, where S and T are finite ...
 5.2.5.2.40: There are 51 houses on a street. Each house has an address between ...
 5.2.5.2.41: Let x be an irrational number. Show that for some positive integer ...
 5.2.5.2.42: Let n" n2 , ... , nl be positive integers. Show that if n, + n2 + ....
 5.2.5.2.43: A proof of Theorem 3 based on the generalized pigeonhole principle ...
Solutions for Chapter 5.2: Counting
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 5.2: Counting
Get Full SolutionsSince 43 problems in chapter 5.2: Counting have been answered, more than 40400 students have viewed full stepbystep solutions from this chapter. Chapter 5.2: Counting includes 43 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to 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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

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 ().

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

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