- 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 3-pe...
- 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
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.
A = CTC = (L.J]))(L.J]))T for positive definite A.
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 A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)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.
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(D» 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.
Skew-symmetric 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)j-I and det V = product of (Xk - Xi) for k > i.