 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 Patricia and is associated to the ISBN: 9780321867322. Since 80 problems in chapter 11.3: Combinations have been answered, more than 19857 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6.

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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!

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

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