- 16-2.1: Show that nC n!(n 2 r)! 3 r!
- 16-2.2: Show that n! 5 n(n 21)!.
- 16-2.3: In 322, evaluate each expression.5!
- 16-2.4: In 322, evaluate each expression.12!
- 16-2.5: In 322, evaluate each expression.8! 4 3!
- 16-2.6: In 322, evaluate each expression.
- 16-2.7: In 322, evaluate each expression.
- 16-2.8: In 322, evaluate each expression.6P6
- 16-2.9: In 322, evaluate each expression.8P4
- 16-2.10: In 322, evaluate each expression.
- 16-2.11: In 322, evaluate each expression.
- 16-2.12: In 322, evaluate each expression.4C3
- 16-2.13: In 322, evaluate each expression.
- 16-2.14: In 322, evaluate each expression.12C5
- 16-2.15: In 322, evaluate each expression.10P4 4 4!
- 16-2.16: In 322, evaluate each expression.
- 16-2.17: In 322, evaluate each expression.Q R 1510 Q R
- 16-2.18: In 322, evaluate each expression.Q44 Q R
- 16-2.19: In 322, evaluate each expression.120 R
- 16-2.20: In 322, evaluate each expression.
- 16-2.21: In 322, evaluate each expression.8C3 4 8C5
- 16-2.22: In 322, evaluate each expression.20C5 4 20C15
- 16-2.23: In 2330, find the number of different arrangements that are possibl...
- 16-2.24: In 2330, find the number of different arrangements that are possibl...
- 16-2.25: In 2330, find the number of different arrangements that are possibl...
- 16-2.26: In 2330, find the number of different arrangements that are possibl...
- 16-2.27: In 2330, find the number of different arrangements that are possibl...
- 16-2.28: In 2330, find the number of different arrangements that are possibl...
- 16-2.29: In 2330, find the number of different arrangements that are possibl...
- 16-2.30: In 2330, find the number of different arrangements that are possibl...
- 16-2.31: A box contains 9 red, 4 blue, and 6 yellow chips. In how many ways ...
- 16-2.32: In 3237, determine the number of different arrangements.The finishi...
- 16-2.33: In 3237, determine the number of different arrangements.Seating of ...
- 16-2.34: In 3237, determine the number of different arrangements.Stacking 6 ...
- 16-2.35: In 3237, determine the number of different arrangements.
- 16-2.36: In 3237, determine the number of different arrangements.Making a ro...
- 16-2.37: In 3237, determine the number of different arrangements.Rating 6 em...
- 16-2.38: In 3843, solve for x.6P2 5 x(5P3
- 16-2.39: In 3843, solve for x.P6 5 30(xP4)
- 16-2.40: In 3843, solve for x.P5 5 1,287(xPx)
- 16-2.41: In 3843, solve for x.xC6 5 xC4
- 16-2.42: In 3843, solve for x.R x8
- 16-2.43: In 3843, solve for x.12C4 5 xC8
- 16-2.44: Show that nCr 5 nCnr .
- 16-2.45: At the library, Jordan selects 8 books that he would like to read b...
- 16-2.46: Eli has homework assignments for 5 subjects but decides to complete...
- 16-2.47: There are 12 boys and 13 girls assigned to a class. In how many way...
- 16-2.48: In how many ways can the letters of CIRCLE be arranged if the first...
- 16-2.49: In how many ways can 6 transfer students be assigned to seats in a ...
- 16-2.50: From a standard deck of 52 cards, how many hands of 5 cards can be ...
- 16-2.51: A local convenience store hires three students to work after school...
- 16-2.52: Marie has an English assignment and a history assignment that requi...
- 16-2.53: Roy and Valerie are playing a game of tic-tac-toe. In how many diff...
- 16-2.54: Rafael is running for mayor of his town. He sent out a survey askin...
- 16-2.55: The Electronic Depot received 108 mp3 players, 12 of which are defe...
- 16-2.56: The program director at the local sports center wants to schedule t...
- 16-2.57: A medical researcher has received approval to test a new combinatio...
Solutions for Chapter 16-2: PERMUTATIONS AND COMBINATIONS
Full solutions for Amsco's Algebra 2 and Trigonometry | 1st Edition
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.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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.
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.
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.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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 .
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
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!
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.