 5.4.5.4.1: Find the expansion of (x + y)4 a) using combinatorial reasoning, as...
 5.4.5.4.2: Find the expansion of (x + y)5 a) using combinatorial reasoning, as...
 5.4.5.4.3: Find the expansion of (x + y)6.
 5.4.5.4.4: Find the coefficient of x5 y8 in (x + y)13.
 5.4.5.4.5: How many terms are there in the expansion of (x + y) IOO after like...
 5.4.5.4.6: What is the coefficient ofx 7 in (I + x) I I?
 5.4.5.4.7: What is the coefficient ofx9 in (2  x)19?
 5.4.5.4.8: What is the coefficient of x8 y9 in the expansion of (3x + 2y)1 7 ?
 5.4.5.4.9: What is the coefficient of x 101 y99 in the expansion of (2x  3y)200?
 5.4.5.4.10: Give a formula for the coefficient of x k in the expansion of (x + ...
 5.4.5.4.11: Give a formula for the coefficient of x k in the expansion of(x2  ...
 5.4.5.4.12: The row of Pascal's triangle containing the binomial coefficients C...
 5.4.5.4.13: What is the row of Pascal's triangle containing the binomial coeffi...
 5.4.5.4.14: Show that if n is a positive integer, then 1 = () < G) < ... < (Ln/...
 5.4.5.4.15: Show that m ::s 2 n for all positive integers n and all integers k ...
 5.4.5.4.16: a) Use Exercise 14 and Corollary 1 to show that if n is an integer ...
 5.4.5.4.17: Show that if n and k are integers with 1 ::s k ::s n, then (;) ::s ...
 5.4.5.4.18: Suppose that b is an integer with b ::::: 7. Use the Binomial Theor...
 5.4.5.4.19: Prove Pascal's Identity, using the formula for C).
 5.4.5.4.20: Suppose that k and n are integers with 1 ::s k < n. Prove the hexag...
 5.4.5.4.21: Prove that if n and k are integers with 1 ::s k ::s n, then kG) = n...
 5.4.5.4.22: Prove the identity C) G) = G) (:::Z), whenever n, r, and k are nonn...
 5.4.5.4.23: Show that if n and k are positive integers, then Use this identity ...
 5.4.5.4.24: Show that if p is a prime and k is an integer such that 1 ::s k ::s...
 5.4.5.4.25: Let n be a positive integer. Show that CI ) + e) = e:12 )/2.
 5.4.5.4.26: Let n and k be integers with 1 ::s k ::s n. Show that L: =I G) ( kl...
 5.4.5.4.27: Prove that whenever n and r are positive integers, a) using a combi...
 5.4.5.4.28: Show that if n is a positive integer, then en = 2(;) + n2 a) using ...
 5.4.5.4.29: Give a combinatorial proof that L:Z = I kG) = n2n  l . [Hint: Coun...
 5.4.5.4.30: Give a combinatorial proof that L:Z = I kG) 2 = ne..=/). [Hint: Co...
 5.4.5.4.31: Show that a nonempty set has the same number of subsets with an odd...
 5.4.5.4.32: Prove the Binomial Theorem using mathematical induction.
 5.4.5.4.33: In this exercise we will count the number of paths in the xy plane ...
 5.4.5.4.34: Use Exercise 33 to prove that G) = tk ) whenever k is an integer wi...
 5.4.5.4.35: Use Exercise 33 to prove Theorem 4. [Hint: Count the number of path...
 5.4.5.4.36: Use Exercise 33 to prove Pascal's Identity. [Hint: Show that a path...
 5.4.5.4.37: Prove the identity in Exercise 27 using Exercise 33. [Hint: First, ...
 5.4.5.4.38: Give a combinatorial proof that if n is a positive integer then L:Z...
 5.4.5.4.39: Determine a formula involving binomial coefficients for the nth ter...
Solutions for Chapter 5.4: Counting
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 5.4: Counting
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Since 39 problems in chapter 5.4: Counting have been answered, more than 40939 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720. Chapter 5.4: Counting includes 39 full stepbystep solutions.

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.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Outer product uv T
= column times row = rank one matrix.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.