 9.2.1: Determine the number of distinct permutations (with repetition) of ...
 9.2.2: Determine the number of distinct arrangements of 9 letters, 3 of wh...
 9.2.3: How many 7digit numbers can be formed from the digits in the follo...
 9.2.4: How many different 8digit numbers can be obtained by permuting the...
 9.2.5: What is the number of 4element multisets whose elements belong to ...
 9.2.6: What is the number of 5element multisets whose elements belong to ...
 9.2.7: A 10question multiple choice quiz is known to have 4 questions whe...
 9.2.8: Eight different computer science books have been purchased to give ...
 9.2.9: Seven young boys have moved into a new community and would like to ...
 9.2.10: For the final exam in a musical theater class, the three students J...
 9.2.11: The Chair of a University Department is making committee assignment...
 9.2.12: Before leaving with his girlfriend to visit her parents, a young ma...
 9.2.13: At a buffet dinner, one of the people at a table (seating 6 people)...
 9.2.14: A professor has 10 identical new pens that he no longer needs. In h...
 9.2.15: Ten people have been selected to receive gift certificates to a res...
 9.2.16: A total of 12 students have volunteered to participate in student g...
 9.2.17: A total of 12 different computer science books are to be given to t...
 9.2.18: A bowl contains a large number of red, blue, green and yellow marbl...
 9.2.19: In how many ways can a man distribute 10 silver dollars to his thre...
 9.2.20: A coin collector has 10 identical silver coins and 6 identical gold...
 9.2.21: In how many ways can 10 identical pens and 7 identical pocket calcu...
 9.2.22: How many 5digit numbers are there, the sum of whose digits is 8?
 9.2.23: How many 3digit numbers are there, the sum of whose digits is 10?
 9.2.24: How many 4digit numbers are there, the sum of whose digits is 11?
 9.2.25: A man enters a bakery just before it closes in order to buy donuts....
 9.2.26: A woman enters Dougs Donuts to purchase some donuts. Give an exampl...
Solutions for Chapter 9.2: Permutations and Combinations with Repetition
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 9.2: Permutations and Combinations with Repetition
Get Full SolutionsChapter 9.2: Permutations and Combinations with Repetition includes 26 full stepbystep solutions. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 26 problems in chapter 9.2: Permutations and Combinations with Repetition have been answered, more than 13003 students have viewed full stepbystep solutions from this chapter.

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det 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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

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

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).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.