 8.6.1: Evaluate. 13C2
 8.6.2: Evaluate. 9C6
 8.6.3: Evaluate.1311b
 8.6.4: Evaluate. a93b
 8.6.5: Evaluate. a71b
 8.6.6: Evaluate. 88b
 8.6.7: Evaluate. 5P33!
 8.6.8: Evaluate. 10P55!
 8.6.9: Evaluate. a60b
 8.6.10: Evaluate. a61b
 8.6.11: Evaluate. a62b
 8.6.12: Evaluate. 63b
 8.6.13: Evaluate. a70b + a71b + a72b + a73b + a74b + a75b+ a76b + a77b
 8.6.14: Evaluate. a60b + a61b + a62b + a63b + a64b+ a65b + a66b
 8.6.15: Evaluate.52C4
 8.6.16: Evaluate. 52C5
 8.6.17: Evaluate. a2711b
 8.6.18: Evaluate. 378b
 8.6.19: Evaluate. an1b
 8.6.20: Evaluate. an3
 8.6.21: Evaluate. ammb
 8.6.22: Evaluate. at4b
 8.6.23: In each of the following exercises, give an expression forthe answe...
 8.6.24: In each of the following exercises, give an expression forthe answe...
 8.6.25: In each of the following exercises, give an expression forthe answe...
 8.6.26: In each of the following exercises, give an expression forthe answe...
 8.6.27: In each of the following exercises, give an expression forthe answe...
 8.6.28: Lines and Triangles from Points. How manylines are determined by 8 ...
 8.6.29: Poker Hands. How many 5card poker handsare possible with a 52card...
 8.6.30: Bridge Hands. How many 13card bridge handsare possible with a 52c...
 8.6.31: BaskinRobbins Ice Cream. Burt Baskin andIrv Robbins began making i...
 8.6.32: Solve. 3x  7 = 5x + 10
 8.6.33: Solve. 2x2  x = 3
 8.6.34: Solve.x2 + 5x + 1 = 0
 8.6.35: Solve. x3 + 3x2  10x = 24
 8.6.36: Full House. A full house in poker consists ofthree of a kind and a ...
 8.6.37: Flush. A flush in poker consists of a 5cardhand with all cards of ...
 8.6.38: There are n points on a circle. How many quadrilateralscan be inscr...
 8.6.39: League Games. How many games are playedin a league with n teams if ...
 8.6.40: Solve for n. an3b = 2 an  12b 41. ann  2b = 6
 8.6.41: Solve for n. Prove thatank  1b + ankb = an + 1kbfor any natural nu...
Solutions for Chapter 8.6: Combinatorics: Combinations
Full solutions for College Algebra: Graphs and Models  5th Edition
ISBN: 9780321783950
Solutions for Chapter 8.6: Combinatorics: Combinations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.6: Combinatorics: Combinations includes 41 full stepbystep solutions. College Algebra: Graphs and Models was written by and is associated to the ISBN: 9780321783950. Since 41 problems in chapter 8.6: Combinatorics: Combinations have been answered, more than 25844 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra: Graphs and Models, edition: 5.

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).

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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

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.

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 .

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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