 9.1: Al has 9 different books. He decides to give 4 to Bob, 3 to Charles...
 9.2: In order to gain access to a private web site, each person must hav...
 9.3: A deck of 12 cards consists of 3 identical red cards, 3 identical b...
 9.4: How many 10digit numbers contain five 1s, three 2s and two 3s?
 9.5: What is the coefficient of x in 2x 1 2x _7 ? Give the exact answer.
 9.6: Expand (a + b)8, giving precise coefficients.
 9.7: What is the exact coefficient of a4b3c2 in (a + b + c)9?
 9.8: (a) Let akak1 . . . a1a0 be the decimal representation of an intege...
 9.9: Determine the number of distinct permutations of the letters of the...
 9.10: Ten students have decided to join a club. If 4 choose the computer ...
 9.11: What is the number of possible birthday months for three people?
 9.12: How many 4digit numbers are there that are divisible by 10 and the...
 9.13: A boy has been saving for a vacation trip and keeps $1 bills, $5 bi...
 9.14: One evening each year, a baseball team has two brothers night, wher...
 9.15: A total of 10 different books are to be given to the top 3 mathemat...
 9.16: A man goes into a donut shop to buy 7 donuts. At the time he is the...
 9.17: A little boy is placing 20 of his marbles in a row. If 12 are red a...
 9.18: How many 4digit numbers are there, the sum of whose digits is 10?
 9.19: A bowl contains 10 balls of each of the colors red, blue and green....
 9.20: A man buys 6 donuts, each of which is a plain donut, a powdered don...
 9.21: Compute the following binomial or extended binomial coefficients: (...
 9.22: Express 1 + x2_1 as a power series.
 9.23: Express (1 3x)2 as a power series.
 9.24: A sequence {an} is defined recursively by a0 = 1, a1 = 2 and an = a...
 9.25: A sequence {an} is defined recursively by a0 = 0, a1 = 1 and an = 2...
 9.26: A sequence {an} is defined recursively by a0 = 0, a1 = 1 and an = 2...
 9.27: At a bookstore, there is a special on childrens books that have bee...
Solutions for Chapter 9: ADVANCED COUNTING METHODS
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 9: ADVANCED COUNTING METHODS
Get Full SolutionsChapter 9: ADVANCED COUNTING METHODS includes 27 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308. Since 27 problems in chapter 9: ADVANCED COUNTING METHODS have been answered, more than 13928 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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 .

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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 .

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

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.

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

Solvable system Ax = b.
The right side b is in the column space of A.

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

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