 8.5.1: A total of 10 prizes of equal value are to be given to 10 graduatin...
 8.5.2: A student committee is to consist of 3 seniors and 4 juniors. A tot...
 8.5.3: An experiment uses three jars A, B and C. Jar A contains five balls...
 8.5.4: From a group of 6 juniors and 5 sophomores, a committee of 3 junior...
 8.5.5: An election will be held to fill three faculty seats and three stud...
 8.5.6: A group of three women and seven men have been nominated for a 3pe...
 8.5.7: From a group of 6 sophomores and 5 freshmen, a committee of 3 sopho...
 8.5.8: A total of 2 freshmen, 3 sophomores, 4 juniors and 5 seniors have b...
 8.5.9: A total of 4 sophomores, 5 juniors and 6 seniors are eligible for 5...
 8.5.10: A total of 5 seniors, 3 juniors and 4 sophomores have volunteered t...
 8.5.11: An investor intends to buy shares of stock in 3 companies, chosen f...
 8.5.12: Give an example of a problem involving students serving on a commit...
 8.5.13: For a positive integer n, a total of 19n students have volunteered ...
Solutions for Chapter 8.5: Applications of Permutations and Combinations
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 8.5: Applications of Permutations and Combinations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 13 problems in chapter 8.5: Applications of Permutations and Combinations have been answered, more than 11956 students have viewed full stepbystep solutions from this chapter. Chapter 8.5: Applications of Permutations and Combinations includes 13 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» 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.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.