 12.1.1: State whether the events are independent or dependent.
 12.1.2: State whether the events are independent or dependent.
 12.1.3: An ice cream shop offers a choice of two types of cones and 15 flav...
 12.1.4: A bookshelf holds 4 different biographies and 5 different mystery n...
 12.1.5: Lances math quiz has eight truefalse questions. How many different...
 12.1.6: Pizza House offers three different crusts, four sizes, and eight to...
 12.1.7: For a college application, Macawi must select one of five topics on...
 12.1.8: State whether the events are independent or dependent.
 12.1.9: State whether the events are independent or dependent.
 12.1.10: Each of six people guess the total number of points scored in a bas...
 12.1.11: The letters A through Z are written on pieces of paper and placed i...
 12.1.12: Tim wants to buy one of three different books he sees in a book sto...
 12.1.13: A video store has 8 new releases this week. Each is available on vi...
 12.1.14: Carlos has homework in math, chemistry, and English. How many ways ...
 12.1.15: The menu for a banquet has a choice of 2 types of salad, 5 main cou...
 12.1.16: A baseball glove manufacturer makes gloves in 4 different sizes, 3 ...
 12.1.17: Each question on a fivequestion multiplechoice quiz has answer ch...
 12.1.18: Abby is registering at a Web site. She must select a password conta...
 12.1.19: How many ways can you arrange the science books?
 12.1.20: Since the science books are to be together, they can be treated lik...
 12.1.21: How many area codes were possible before 1995?
 12.1.22: In 1995, the restriction on the middle digit was removed, allowing ...
 12.1.23: How many ways can six different books be arranged on a shelf if one...
 12.1.24: In how many orders can eight actors be listed in the opening credit...
 12.1.25: How many different 5digit codes are possible using the keypad show...
 12.1.26: Use the Internet or other resource to find the configuration of let...
 12.1.27: Describe a situation in which you can use the Fundamental Counting ...
 12.1.28: Explain how choosing to buy a car or a pickup truck and then select...
 12.1.29: The members of the Math Club need to elect a president and a vice p...
 12.1.30: Use the information on page 684 to explain how you can count the ma...
 12.1.31: How many numbers between 100 and 999, inclusive, have 7 in the tens...
 12.1.32: A coin is tossed four times. How many possible sequences of heads o...
 12.1.33: Prove that 4 + 7 + 10 + + (3n + 1) = _ n(3n + 5) 2 for all positive...
 12.1.34: Find the indicated term of each expansion.
 12.1.35: Find the indicated term of each expansion.
 12.1.36: Edison is located at (9, 3) in the coordinate system on a road map....
 12.1.37: Solve each equation by factoring
 12.1.38: Solve each equation by factoring
 12.1.39: Solve each equation by factoring
 12.1.40: Solve each matrix equation
 12.1.41: Solve each matrix equation
 12.1.42: Evaluate each expression.
 12.1.43: Evaluate each expression.
 12.1.44: Evaluate each expression.
 12.1.45: Evaluate each expression.
 12.1.46: Evaluate each expression.
 12.1.47: Evaluate each expression.
 12.1.48: Evaluate each expression.
 12.1.49: Evaluate each expression.
Solutions for Chapter 12.1: The Counting Principle
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 12.1: The Counting Principle
Get Full SolutionsSince 49 problems in chapter 12.1: The Counting Principle have been answered, more than 44241 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 12.1: The Counting Principle includes 49 full stepbystep solutions. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

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

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].

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.

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

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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!).

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

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

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.

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