Discreet Math Question:

Suppose you have 100 students in school. Principal wants to take photos in different ways so that every student thinks they have been treated fairly.

1.She wants to divide 100 students into groups of 10 and take one photo of each group. So no student is in more than one photo and she doesn't care how they are lined up. How many photo are going to need to be taken in this situation?

2. She has a new idea that includes three of the schools teachers. She still wants to divide the 100 students into groups of 10. With each student, they will take a picture with every possible way to choose two of the three teachers, with one on the left and other on the right (So Order Matters). How many photos are going to be needed in this situation?

Let S be the set of all codes over Fq of length n.LetC1, C2∈ S.DefineC1∼C2to mean that there exists an n×n monomial matrix M such that C1=M C2.Probe that∼is an equivalence relation on S.LetC1, C2such that C1∼C2. Do C1 and C2have the same dimension?,length?, minimum distance?, if C1is self-orthogonal, is also C1 self-orthogonal?.

V is a subspace of inner-product space R³, generated by vector

u =[1 1 2]^T and v =[ 2 2 3]^T.

T is transpose

(1) Find its orthogonal complement space V^┴ ;

(2) Find the dimension of space W = V+ V^┴;

(3) Find the angle q between u and v; also the angle b between u and normalized x with respect to its 2-norm.

(4) Considering v’ = av, a is a scaler, show the angle q’ between u and v’

1.) How many relations are there from a set of size n to a set of size m?

2.) Determine the number of entries in the following sequences:

a.) {13, 19, 25, . . . , 601}

b. {7, 11, 19, 35, 67, . . . , 131075}

Let G be a nonabelian group of order 253=23(11), let P<G be a Sylow 23-subgroup and Q<G a Sylow 11-subgroup.

a. What are the orders of P and Q. (Explain and include any theorems used).

b. How many distinct conjugates of P and Q are there in G? n23? n11? (Explain, include any theorems used).

c. Prove that G is isomorphic to the semidirect product of P and Q.

(a) Find the equilibrium solution, or critical point, of the given system.

(b) Use a computer to draw a direction field and phase portrait centered at the critical point.

(c) Describe how solutions of the system behave in the vicinity of the critical point.

x′ =−0.25x−0.75y+8, y′ =0.5x+y−11.5

(d) Let x= x_c+u and y= y_c+v, where x_c and y_c give the critical point you found in (a). Plug these into the system and show that you obtain a homogeneous system u′ = Au for u = (u v)^T .

(e) Solve the resulting homogeneous system for u and v, and show that the solutions you obtain match the phase portrait that you generated in (b).

Q1. The following table shows the quantity supplied and quantity demanded of a commodity at certain unit prices.

Unit Price
$1.50
1150
340

Quantity Demanded
$2.75
712.5
652.5

Quantity Supplied
$3.25
537.5
777.5

a. From the information given in the table above, describe in your own words, how would you go about showing that the quantity demanded and supplied are linear functions of price, without having to plot a graph.

b. By demonstrating what you have describe above, show that the quantity demanded and supplied are linear functions of price and that price is equal to $2.85 when quantity demanded is equal to quantity supplied.

Q4. A fish processing company in Charlotteville processes three types of products: Type A, Type B and Type C. Its operations involve three basic activities: cleaning, cutting and packaging. Type A products require 4 minutes of cleaning 2 minutes of cutting and 2 minutes of packaging time. Type B products require 6 minutes of cleaning 4 minutes of cutting and 2 minutes of packaging time. Type C products require 8 minutes of cleaning 2 minutes of cutting and 4 minutes of packaging time.

Given that the total time available for cleaning, cutting and packaging is 3.5hours, 2.5 hours and 1.5 hours respectively.

a. Clearly defining your variables, show the model which represents the information given in this problem. State any restrictions on the variables in the problem.

Ten students shall be sorted into the four houses (namely Gryffindor, Slytherin, Ravenclaw, and Hufflepuff) in Hogwarts. If at least one student shall go to each house, how many ways can the students be sorted?

in how many ways can you distribute 13 identical pieces of candy to 5 children, if two of the children are twins and must get an equal number of pieces and every child must receive at least one piece?

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]_p, [2]_p, . . . , [p-1]_p}. Prove that for y ≠ 0, L_y restricts to a bijective map L_y|_s : S → S.

11.5 Prove Fermat's Little Theorem

solve the initial value problem

Y" + 2Y' - Y = 0, Y(0)=0,Y'(0) = 2sqrt2

With regards to elliptic curves, what is the point at infinity? Consider cases where x and y vary over real numbers.

In Elliptic Curves, when computing A⨁B = C , we take the line through A and B and find the point it intersects the curve. We then reflect through the x-axis. Why do we reflect at the x-axis?