Suppose that we put 4 rooks on a standard 8 × 8 chess board so that none of the rooks can capture the others. This means that no two rooks can appear in the same row or column. Furthermore, suppose that we do not put a rook in the upper left corner. How many ways can we do this?

Prove that all rotations and translations form a subgroup of the group of all reflections and products of reflections in Euclidean Geometry. What theorems do we use to show that this is a subgroup?

I know that I need to show that the subset is

closed

identity is in the subset

every element in the subset has an inverse in the subset.

I don't have to prove associative property since that is already proven with Isometries. What theorems for rotations and translations so that they are closed, identity is in the subset and every element is the subset has an inverse in the subset.

The following four methods are commonly used for solving systems of equations:
1.    Graphing
2.    Substitution
3.    Addition
4.    Determinants and Cramer’s Rule

Pick one method and discuss the pros and cons of that method. Provide an example of a problem that can be easily solved using your chosen method and an example of a problem that would be more difficult to solve using your method. Review your classmates’ responses to find a classmate who chose a different method. Discuss that alternative method with your classmate.

Prove that if ∑a_n converges absolutely, then both ∑P_n and ∑N_n converge

1a. Find the number of ways to rearrange each of the following words a. GUIDE b. SCHOOL c. SALESPERSONS

1b. A handful of 6 jellybeans is drawn from a jar that contains 5 different flavors: blueberry, popcorn, pineapple, apple, lemon. a. What outcome does × × ×| × | | × ×| represent? b. How many ways are there to select a handful of 6 jellybeans from the jar?

1c. How many integer solutions are there to the equation x + y + z = 8 where x, y, and z are all greater than or equal to zero?

1d. How many ways are there to choose a dozen donuts from the 21 varieties at a donut shop?

7.              Determine the first 4 nonzero terms of the Taylor series for the solution y = φ(x) of the given initial value problem, y^’’ + cos(x)y^’ + x²y = 0; y(0) = 1, y^’(0) = 1.

What do you expect the radius of convergence to be? Why?

please show all steps

A given field mouse population satisfies the differential equation

d⁢p/d⁢t=0.2p-390.

where p is the number of mice and t is the time in months.

(a) Find the time at which the population becomes extinct if

   p(0)=1940⁢   .

Round your answer to two decimal places.

tf=

(c) Find the initial population p0 if the population is to become extinct in 1 year.

Round your answer to the nearest integer.

 p0⁢= 

This is a linear algebra question.

Determine whether the given system has a unique solution, no solution, or infinitely many solutions. Put the associated augmented matrix in reduced row echelon form and find solutions, if any, in vector form. (If the system has infinitely many solutions, enter a general solution in terms of s. If the system has no solution, enter NO SOLUTION in any cell of the vector.)

Given a natural number q ≥ 1, define a relation ∼ on the set Z by x ∼ y
if x - y is divisible by q.
(i) Show that ∼ is an equivalence relation.
We will denote the set of equivalence classes defined by ∼ with Z=qZ. Also
let x mod q denote the equivalence class to which an integer x belongs.
(ii) Check that the operations

are well-defined on Z=qZ.
(iii) With the operations as defined above, show that Z=qZ is not a field
if q is not prime.

3. How many strings can be made using 9 or more letters of MISSISSIPPI?

find the order of 3 modulo 29

The Carolina Cougars is a major league baseball expansion team beginning its third year of operation. The team had losing records in each of its first 2 years and finished near the bottom of its division. However, the team was young and generally competitive. The team’s general manager, Frank Lane, and manager, Biff Diamond, believe that with a few additional good players, the Cougars can become a contender for the division title and perhaps even for the pennant. They have prepared several proposals for free- agent acquisitions to present to the team’s owner, Bruce Wayne.

Under one proposal the team would sign several good available free agents, including two pitchers, a good fielding shortstop, and two power-hitting outfielders for $52 million in bonuses and annual salary. The second proposal is less ambitious, costing $20 million to sign a relief pitcher, a solid, good-hitting infielder, and one power-hitting out- fielder. The final proposal would be to stand pat with the current team and continue to develop.

General Manager Lane wants to lay out a possible season scenario for the owner so he can assess the long-run ramifications of each decision strategy. Because the only thing the owner understands is money, Frank wants this analysis to be quantitative, indicating the money to be made or lost from each strategy. To help develop this analysis, Frank has hired his kids, Penny and Nathan, both management science graduates from Tech.

Penny and Nathan analyzed league data for the previous five seasons for attendance trends, logo sales (i.e., clothing, souvenirs, hats, etc.), player sales and trades, and revenues. In addition, they interviewed several other owners, general managers, and league officials. They also analyzed the free agents that the team was considering signing.

Based on their analysis, Penny and Nathan feel that if the Cougars do not invest in any free agents, the team will have a 25% chance of contending for the division title and a 75% chance of being out of contention most of the sea- son. If the team is a contender, there is a .70 probability that attendance will increase as the season progresses and the team will have high attendance levels (between 1.5 million and 2.0 million) with profits of $170 million from ticket sales, concessions, advertising sales, TV and radio sales, and logo sales. They estimate a .25 probability that the team’s attendance will be mediocre (between 1.0 million and 1.5 million) with profits of $115 million and a .05 prob- ability that the team will suffer low attendance (less than 1.0 million) with profit of $90 million. If the team is not a contender, Penny and Nathan estimate that there is .05 probability of high attendance with profits of $95 mil- lion, a .20 probability of medium attendance with profits of $55 million, and a .75 probability of low attendance with profits of $30 million.

If the team marginally invests in free agents at a cost of $20 million, there is a 50–50 chance it will be a contender. If it is a contender, then later in the season it can either stand pat with its existing roster or buy or trade for players that could improve the team’s chances of winning the division. If the team stands pat, there is a .75 probability that attendance will be high and profits will be $195 million. There is a .20 probability that attendance will be mediocre with profits of $160 million and a .05 probability of low attendance and profits of $120 million. Alternatively, if the team decides to buy or trade for players, it will cost $8 million, and the probability of high attendance with profits of $200 million will be .80. The probability of mediocre attendance with $170 million in profits will be .15, and there will be a .05 probability of low attendance, with profits of $125 million.

If the team is not in contention, then it will either stand pat or sell some of its players, earning approximately $8 million in profit. If the team stands pat, there is a .12 probability of high attendance, with profits of $110 million; a .28 probability of mediocre attendance, with profits of $65 million; and a .60 probability of low attendance, with profits of $40 million. If the team sells players, the fans will likely lose interest at an even faster rate, and the probability of high attendance with profits of $100 million will drop to .08, the probability of mediocre attendance with profits of $60 million will be .22, and the probability of low attendance with profits of $35 million will be .70.

The most ambitious free-agent strategy will increase the team’s chances of being a contender to 65%. This strategy will also excite the fans most during the off-season and boost ticket sales and advertising and logo sales early in the year. If the team does contend for the division title, then later in the season it will have to decide whether to invest in more players. If the Cougars stand pat, the probability of high attendance with profits of $210 million will be .80, the probability of mediocre attendance with profits of $170 million will be .15, and the probability of low attendance with profits of $125 million will be .05. If the team buys players at a cost of $10 million, then the probability of having high attendance with profits of $220 million will increase to .83, the probability of mediocre attendance with profits of $175 million will be .12, and the probability of low attendance with profits of $130 million will be .05.

If the team is not in contention, it will either sell some players’ contracts later in the season for profits of around $12 million or stand pat. If it stays with its roster, the prob- ability of high attendance with profits of $110 million will be .15, the probability of mediocre attendance with profits of $70 million will be .30, and the probability of low attendance with profits of $50 million will be .55. If the team sells players late in the season, there will be a .10 probability of high attendance with profits of $105 million, a .30 probability of mediocre attendance with profits of $65 mil- lion, and a .60 probability of low attendance with profits of $45 million.

Assist Penny and Nathan in determining the best strategy to follow and its expected value.

Set up the addition and multiplication tables for Z3 and Z6.
Use these tables to verify that (Z3, +), (Z3 \ {0}, ·) and (Z6, +) are groups, but (Z6 \ {0}, ·) is not a group.

In which finite field does "25 divided by 5 is 14" hold?

Let G be a graph. prove G has a Eulerian trail if and only if G has at most one non-trivial component and at most 2 vertices have odd degree