 6.R.1RQ: Explain how the sum and product rules can be used to find the numbe...
 6.R.2RQ: Explain how to find the number of bit strings of length not exceedi...
 6.R.3RQ: a) How can the product rule be used to find the number of functions...
 6.R.5RQ: How can you find the number of bit strings of length ten that eithe...
 6.R.4RQ: How can you find the number of possible outcomes of a playoff betwe...
 6.R.6RQ: a) State the pigeonhole principle.________________b) Explain how th...
 6.R.7RQ: a) State the generalized pigeonhole principle.________________b) Ex...
 6.R.8RQ: a) What is the difference between an rcombination and an rpermuta...
 6.R.10RQ: What is meant by a combinatorial proof of an identity? How is such ...
 6.R.9RQ: a) What is Pascal’s triangle?________________b) How can a row of Pa...
 6.R.12RQ: a) State the binomial theorem.________________b) Explain how to pro...
 6.R.11RQ: Explain how to prove Pascal's identity using a combinatorial argument.
 6.R.13RQ: a) Explain how to find a formula for the number of ways to select r...
 6.R.15RQ: a) Derive a formula for the number of permutations of n objects of ...
 6.R.14RQ: a) Let n and r be positive integers. Explain why the number of solu...
 6.R.16RQ: Describe an algorithm for generating all the permutations of the se...
 6.R.17RQ: a) How many ways are there to deal hands of five cards to six playe...
 6.R.18RQ: Describe an algorithm for generating all the combinations of the se...
Solutions for Chapter 6.R: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 6.R
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Chapter 6.R includes 18 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 18 problems in chapter 6.R have been answered, more than 188948 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

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