 11.3.1: In 14, does the problem involve permutations or combinations? Expla...
 11.3.2: In 14, does the problem involve permutations or combinations? Expla...
 11.3.3: In 14, does the problem involve permutations or combinations? Expla...
 11.3.4: In 14, does the problem involve permutations or combinations? Expla...
 11.3.5: In 520, use the formula for nCr to evaluate each expression.6C5
 11.3.6: In 520, use the formula for nCr to evaluate each expression.. 8C7
 11.3.7: In 520, use the formula for nCr to evaluate each expression.. 9C5
 11.3.8: In 520, use the formula for nCr to evaluate each expression.10C6
 11.3.9: In 520, use the formula for nCr to evaluate each expression.11C4
 11.3.10: In 520, use the formula for nCr to evaluate each expression.12C5
 11.3.11: In 520, use the formula for nCr to evaluate each expression.. 8C1
 11.3.12: In 520, use the formula for nCr to evaluate each expression.7C1
 11.3.13: In 520, use the formula for nCr to evaluate each expression.. 7C7
 11.3.14: In 520, use the formula for nCr to evaluate each expression.C4
 11.3.15: In 520, use the formula for nCr to evaluate each expression.30C3
 11.3.16: In 520, use the formula for nCr to evaluate each expression.25C4
 11.3.17: In 520, use the formula for nCr to evaluate each expression.5C0
 11.3.18: In 520, use the formula for nCr to evaluate each expression.6C0
 11.3.19: In 520, use the formula for nCr to evaluate each expression.7C35C4
 11.3.20: In 520, use the formula for nCr to evaluate each expression.10C36C4
 11.3.21: In 2128, evaluate each expression.
 11.3.22: In 2128, evaluate each expression.
 11.3.23: In 2128, evaluate each expression.
 11.3.24: In 2128, evaluate each expression.
 11.3.25: In 2128, evaluate each expression.
 11.3.26: In 2128, evaluate each expression.
 11.3.27: In 2128, evaluate each expression.
 11.3.28: In 2128, evaluate each expression.
 11.3.29: Use the formula for nCr to solve 2940.An election ballot asks voter...
 11.3.30: Use the formula for nCr to solve 2940.A fourperson committee is to...
 11.3.31: Use the formula for nCr to solve 2940.Of 12 possible books, you pla...
 11.3.32: Use the formula for nCr to solve 2940.There are 14 standbys who hop...
 11.3.33: Use the formula for nCr to solve 2940.You volunteer to help drive c...
 11.3.34: Use the formula for nCr to solve 2940.Of the 100 people in the U.S....
 11.3.35: Use the formula for nCr to solve 2940.To win at LOTTO in the state ...
 11.3.36: Use the formula for nCr to solve 2940.To win in the New York State ...
 11.3.37: Use the formula for nCr to solve 2940.In how many ways can a commit...
 11.3.38: Use the formula for nCr to solve 2940.How many different committees...
 11.3.39: Use the formula for nCr to solve 2940.The U.S. Senate of the 109th ...
 11.3.40: Use the formula for nCr to solve 2940.A mathematics exam consists o...
 11.3.41: In 4160, solve by the method of your choice.In a race in which six ...
 11.3.42: In 4160, solve by the method of your choice.A book club offers a ch...
 11.3.43: In 4160, solve by the method of your choice.A medical researcher ne...
 11.3.44: In 4160, solve by the method of your choice.Fifty people purchase r...
 11.3.45: In 4160, solve by the method of your choice.From a club of 20 peopl...
 11.3.46: In 4160, solve by the method of your choice.Fifty people purchase r...
 11.3.47: In 4160, solve by the method of your choice.How many different four...
 11.3.48: In 4160, solve by the method of your choice.Nine comedy acts will p...
 11.3.49: In 4160, solve by the method of your choice.Using 15 flavors of ice...
 11.3.50: In 4160, solve by the method of your choice.BaskinRobbins offers 3...
 11.3.51: In 4160, solve by the method of your choice.. A restaurant lunch sp...
 11.3.52: In 4160, solve by the method of your choice.There are six employees...
 11.3.53: In 4160, solve by the method of your choice.You have three dress sh...
 11.3.54: In 4160, solve by the method of your choice.You have four flannel s...
 11.3.55: In 4160, solve by the method of your choice.A chef has five brands ...
 11.3.56: In 4160, solve by the method of your choice.In the Mathematics Depa...
 11.3.57: In 4160, solve by the method of your choice.Three are four Democrat...
 11.3.58: In 4160, solve by the method of your choice.An office employs six c...
 11.3.59: In 4160, solve by the method of your choice.A group of campers is g...
 11.3.60: In 4160, solve by the method of your choice.Your mom and dad have d...
 11.3.61: Thousands of jokes have been told about marriage and divorce. 6168 ...
 11.3.62: Thousands of jokes have been told about marriage and divorce. 6168 ...
 11.3.63: Thousands of jokes have been told about marriage and divorce. 6168 ...
 11.3.64: Thousands of jokes have been told about marriage and divorce. 6168 ...
 11.3.65: Thousands of jokes have been told about marriage and divorce. 6168 ...
 11.3.66: Thousands of jokes have been told about marriage and divorce. 6168 ...
 11.3.67: Thousands of jokes have been told about marriage and divorce. 6168 ...
 11.3.68: Thousands of jokes have been told about marriage and divorce. 6168 ...
 11.3.69: What is a combination?
 11.3.70: Explain how to distinguish between permutation and combination prob...
 11.3.71: Write a word problem that can be solved by evaluating 7C3.
 11.3.72: Make Sense? In 7275, determine whether each statement makes sense o...
 11.3.73: Make Sense? In 7275, determine whether each statement makes sense o...
 11.3.74: Make Sense? In 7275, determine whether each statement makes sense o...
 11.3.75: Make Sense? In 7275, determine whether each statement makes sense o...
 11.3.76: Write a word problem that can be solved by evaluating 10C3 # 7C2.
 11.3.77: A 6/53 lottery involves choosing 6 of the numbers from 1 through 53...
 11.3.78: If the number of permutations of n objects taken r at a time is six...
 11.3.79: In a group of 20 people, how long will it take each person to shake...
 11.3.80: A sample of 4 telephones is selected from a shipment of 20 phones. ...
Solutions for Chapter 11.3: Combinations
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter 11.3: Combinations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 11.3: Combinations includes 80 full stepbystep solutions. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. Since 80 problems in chapter 11.3: Combinations have been answered, more than 51220 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

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

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

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

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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.

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