Let P = (p1,...,pn) be a permutation of [n]. We say a number i is a fixed point of p, if pi = i.
(a) Determine the number of permutations of [6] with at most three fixed points.
(b) Determine the number of 9-derangements of [9] so that each even number is in an even position.
(c) Use the following relationship (not proven here, but relatively easy to see) for the Rencontre numbers:
Dn =(n-1)-(Dn-1 +Dn-2) (∗)
to perform an alternative proof of theorem 2.7. So, with the help of (∗), show that for all n ∈ N applies: n Dn =n! r=0 (-1)r r!
(Note: Of course, do not use Sentence 2.7 or Corollary 2.2, it is D0 = 1 and D1 = 0. Note that (∗) is also valid for n = 1 because of the factor (n - 1), no matter how we would define D-1. Then first look at the numbers An = Dn-nDn-1 (∗∗) and show that An = (-1)n is valid. Then divide both sides of (∗∗) by n! and deduce from this the assertion).

y'' - 8y = 4, y(1) = 9, y'(0) = 5

3) If you have a 40% probability of winning at a game of roulette, how many games can you expect to win after playing 30 games?

4) Calculate the variance of the problem above.

5) If Sarah rolls a 6-sided number cube, how many times can she expect to roll a 4 if she plays 18 games?

Find the general solution to the equation

?″+4?=1

this is abstract algebra

Let M be a Q[x]-module which is finite-dimensional as a vector space. What is its torsion submodule?

give an example to show it is false or argue why it is true.

∃!xP(x)⇒∃xP(x)

∃xP(x)⇒∃!xP(x)

∃!x¬P(x)⇒¬∀xP(x)

using the method of Characteristics, Explain the Process for solving the General Advection Equation

this is abstract algebra

Let M be a Q[x]-module which is finite-dimensional as a vector space. What is its torsion submodule?

Find the kernel (or nullspace) and the image of each linear map together with their basis.

(a) T :R4 →R3 given by f(x,y,z,w)=(3x+y−3z+3w , x+y+z+w , 2x+y−z+2w)

(b) T :R3 →R5 given by f(x,y,z)=(2x−y+6z , x−y−z , x+y−5z , z−y , −x+2z)

c) T :R4 →R3 given by f(x,y,z,w)=(x−y−3z+w , 2x−3y+z+2w , 3x+y−4z−w)

Captain John’s Yachts, Inc., located in Fort Lauderdale, Florida, rents three types of ocean-going boats: sailboats, cabin cruisers, and Captain John’s favorite, the luxury yachts. Captain John advertises his boats with his famous “you rent—we pilot” slogan, which means that the company supplies the captain and crew for each rented boat. Each rented boat has one captain, of course, but the crew sizes (deck hands, galley hands, etc.) differ. The crew requirements, in addition to a captain, are one for sailboats, two for cabin cruisers, and three for yachts. Ten employees are captains, and an additional 18 employees fill the various crew positions. Currently, Captain John has rental requests for all his boats: four sailboats, eight cabin cruisers, and three luxury yachts. If Captain John’s daily profit contribution is $50 for sailboats, $70 for cruisers, and $100 for luxury yachts, how many boats of each type should he rent? Prepare and solve.

Industrial design has been awarded a contract to design a label for a new wine produced by Lake View Winery.The company estimates that150 hours will be required to complete the project.The firm's three graphic designers available for to this project are Lisa,a senior designer and team leader; David , a senior designer; and Sarah, a junior designer. Because Lisa has worked on several projects for Lake View Winery, management has specified that Lisa must be assigned at least 40%of the total number of hours assigned to the two senior designers. To provide label -designing experience for Sarah, she must be assigned at least 15% of the total project time. However, the number of hours assigned to Sarah. must not exceed 25% of the total number of hours assigned to the two senior designers.   Due to other commitments, Lisa has a maximum of 50 hours available to work on this project. Hourly wage rates are $30 for Lisa, $25 for David, and $18 for Sarah.

a. Formulate a linear program that can be used to determine the number of hours each graphic designer should be assigned to the project in order to minimize the total cost.

b. How many hours should each graphic designer be assigned to the project?

c. Suppose Lisa could be assigned more than 50 hours. What effect would this have on the optimal solution? Explain.

d. If Sarah were not required to work a minimum number of hours on this project, would the optimal solution change? Explain.

I am having the most trouble with 1d:

1. a. Prove that if f : A → B, g : B → C, and g ◦f : A → C is a 1-to-1 surjection, then f is 1-to-1 and g is a surjection.
Proof.
b. Prove that if f : A → B, g : B → C, g ◦ f : A → C is a 1-to-1 surjection, and g is 1-to-1, then f is a surjection.
Proof.
c. Prove that if f : A → B, g : B → C, g ◦ f : A → C is a 1-to-1 surjection, and f is a surjection, then g is 1-to-1.
Proof.
d. Find functions f : R → R and g : R → R such that g◦f is a 1-to-1 surjection , f is not a surjection, and g is not 1-to-1.