In each case, determine the number of ways

(a) 10 identical candies must be distributed among 4 children

(b) A 15-letter sequences must be made up of 5 A's, 5 B's and 5 C's

(c) 10 identical rings must be placed on your 10 fingers

(d) 3 red, 3 green and 3 blue flags are to be arranged along the street for the parade

The question is: Let G be a finite group, H, K be normal subgroups of G, and H∩K is also a normal subgroup of G. Using Homomorphism theorem ( or First Isomorphism theorem) prove that G/(H∩K) is isomorphism to a subgroup of (G/H)×(G/K). And give a example of group G with normal subgroups H and K such that G/(H∩K) ≆ (G/H)×(G/K), with explanation.

I was trying to find some solutions for the isomorphism proof part, but they all seems to have the condition with H∩K = {e} . I can ensure that there is no missing condition in my question. As there is another subquestion which I've already know the solution, is about given H∩K = {e} and show G/(H∩K) is isomorphism to (G/H)×(G/K).

Soundex produces two models of satellite radios. Model A requires 15 minutes of work on Assembly Line I and 10 minutes of work on Assembly Line II. Model B requires 10 minutes of work on Assembly Line I and 12 minutes of work on Assembly Line II. At most 25 hours of assembly time on Line I and 22 hours of assembly time on Line II are available each day. Soundex anticipates a profit of $12 on Model A and $10 on Model B. Because of previous overproduction, management decides to limit the production of Model A satellite radios to no more than 80 per day.

1. Find the range of values that the resource associated with the time constraint on Assembly Line I can

   assume.

Find the solution of the following initial value problem.

y''' + y'' + y' + y = e^-t + 4cost ; y(0)= 0, y'(0)= -1, y''(0)= 0

Every teacher at a high school is given a n-digit code, e.g. 530...297 (n digits in total).

(1) How many different codes are there?

(2) How many codes read the same backward and forward? (Consider the cases where n is odd or even.)

(3) How many codes contain odd digits only?

(4) How many codes contain at least one even digit?

(5) Consider the case where n is 6. How many codes have distinct digits? (That is, no digit appears more than once.)

Gram method for computing least squares approximate solution. Algorithm 12.1 in the textbook uses the QR factorization to compute the least squares approximate solution xˆ = A†b, where the m × n matrix A has linearly independent columns. It has a complexity of 2mn2 flops. In this exercise we consider an alternative method: First, form the Gram matrix G = AT A and the vector h = AT b; and then compute xˆ = G−1h (using algorithm 11.2 in the textbook). What is the complexity of this method? Compare it to algorithm 12.1. Remark. You might find that the Gram algorithm appears to be a bit faster than the QR method, but the factor is not large enough to have any practical significance. The idea is useful in situations where G is partially available and can be computed more efficiently than by multiplying A and its transpose.

Use the Euclidean algorithm to find the GCD of 3 + 9i and 7-i

6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f = {(1, 5),(2, 5),(3, 6),(x, y)} where x ∈ A and y ∈ B are to be determined by you. (a) In how many ways can you pick x ∈ A and y ∈ B such that f is not a function? (b) In how many ways can you pick x ∈ A and y ∈ B such that f : A → B is onto? (c) In how many ways can you pick x ∈ A and y ∈ B such that f : A → B is not

Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation

w'' - 6x² w' + w = 0

Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation

(x² + 7)y'' + y = 0

A) A 50 gallon tank initially contains 10 gallons of fresh water. At t = 0 t = 0 a brine solution containing 1 pound of salt per gallon is poured into the tank at the rate of 4 gal/min., while the well-stirred mixture leaves the tank at the rate of 1 gal/min. Find the amount of salt in the tank at the moment of overflow.

B) A tank contains 100100 g of salt and 400400 L of water. Water that contains 1414 grams of salt per liter enters the tank at the rate 44 L/min. The solution is mixed and drains from the tank at the rate 66 L/min.

Let yy be the number of g of salt in the tank after tt minutes.

The differential equation for this situation would be:

dydt=dydt=     

Given the initial condition y(0)y(0) = 100 The particular solution would be

y(t)y(t)=    

K-TAB, a slow-release potassium chloride tablet, contains 750 mg of potassium chloride [KCl; MW = 74.5] in a wax/polymer matrix. How many milliequivalents of potassium chloride (round to the nearest whole number) are supplied by a 1 tablet dose given 3 times a day?

Manager T. C. Downs of Plum Engines, a producer of lawn mowers and leaf blowers, must develop an aggregate plan given the forecast for engine demand shown in the table. The department has a regular output capacity of 130 engines per month. Regular output has a cost of $60 per engine. The beginning inventory is zero engines. Overtime has a cost of $90 per engine.

Explain the geometric interpretation of exact differential equations. Talk about gradients, the multivariable chain rule, parametric curves and velocity or tangent vectors. What do these have to do with the condition for exactness of a differential equation? Use a specific example and draw pictures to elaborate.