 121.1: choosing the color and size of a pair of shoes
 121.2: choosing the winner and runnerup at a dog show
 121.3: An ice cream shop offers a choice of two types of cones and 15 flav...
 121.4: STANDARDIZED TEST PRACTICE A bookshelf holds 4 different biographie...
 121.5: Lances math quiz has eight truefalse questions. How many different...
 121.6: Pizza House offers three different crusts, four sizes, and eight to...
 121.7: For a college application, Macawi must select one of five topics on...
 121.8: choosing a president, vicepresident, secretary, and treasurer for ...
 121.9: selecting a fiction book and a nonfiction book at the library
 121.10: Each of six people guess the total number of points scored in a bas...
 121.11: The letters A through Z are written on pieces of paper and placed i...
 121.12: Tim wants to buy one of three different books he sees in a book sto...
 121.13: A video store has 8 new releases this week. Each is available on vi...
 121.14: Carlos has homework in math, chemistry, and English. How many ways ...
 121.15: The menu for a banquet has a choice of 2 types of salad, 5 main cou...
 121.16: A baseball glove manufacturer makes gloves in 4 different sizes, 3 ...
 121.17: Each question on a fivequestion multiplechoice quiz has answer ch...
 121.18: PASSWORDS Abby is registering at a Web site. She must select a pass...
 121.19: How many ways can you arrange the science books?
 121.20: Since the science books are to be together, they can be treated lik...
 121.21: How many area codes were possible before 1995?
 121.22: In 1995, the restriction on the middle digit was removed, allowing ...
 121.23: How many ways can six different books be arranged on a shelf if one...
 121.24: In how many orders can eight actors be listed in the opening credit...
 121.25: HOME SECURITY How many different 5digit codes are possible using t...
 121.26: RESEARCH Use the Internet or other resource to find the configurati...
 121.27: OPEN ENDED Describe a situation in which you can use the Fundamenta...
 121.28: REASONING Explain how choosing to buy a car or a pickup truck and t...
 121.29: CHALLENGE The members of the Math Club need to elect a president an...
 121.30: Writing in Math Use the information on page 684 to explain how you ...
 121.31: ACT/SAT How many numbers between 100 and 999, inclusive, have 7 in ...
 121.32: REVIEW A coin is tossed four times. How many possible sequences of ...
 121.33: Prove that 4 + 7 + 10 + + (3n + 1) = _ n(3n + 5) 2 for all positive...
 121.34: third term of (x + y )
 121.35: fifth term of (2a  b )
 121.36: CARTOGRAPHY Edison is located at (9, 3) in the coordinate system on...
 121.37: x 2  16 = 0
 121.38: x 2  3x  10 = 0
 121.39: 3 x 2 + 8x  3 = 0
 121.40: [x y] = [y 4]
 121.41: 3y 2x = x + 8 y  x
 121.42: 5! 2!
 121.43: 6! 4!
 121.44: 7! 3!
 121.45: 6! 1!
 121.46: 4! 2!2!
 121.47: 6! 2!4!
 121.48: 8! 3!5!
 121.49: 5! 5!0!
Solutions for Chapter 121: The Counting Principle
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 121: The Counting Principle
Get Full SolutionsSince 49 problems in chapter 121: The Counting Principle have been answered, more than 60812 students have viewed full stepbystep solutions from this chapter. Chapter 121: The Counting Principle includes 49 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

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 column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.