find the number of ways to distribute 18 balls, three each of six different colors, into three boxes

1. Consider the set of real numbers S=[-1,0)U{1/n:n=1,2,3,...}. Give the explicit set for each of the

1. The complement of S,
2. The closure of S,
3. The boundary points of S,
4. The limit points of S,
5. The isolated points of S,
6. The interior of S, and
7. The exterior of S.

Explain in simple terms what a primitive root is for prime modulus, and show that 2 is a primitive root of 11, but 3 is not.

Find the series solution of the following equation x(d^2y/dx^2+(1-x)dy/dx+ny=0. where n=0,1,2

According to Zeller's Formula, the day of the week can be calculated by:

f=k+(13m−1)/5+D+D/4+C/4−2C

where:

•k is the day of the month.

•m is the month number designated in a special way: March is 1, April is 2, . . . , December is10; January is 11, and February is 12. If x is the usual month number, i.e. for January x is 1, for February x is 2, and so on; then m can be computed with this formula:m= (x+21)%12+1,where % is the usual modulus (i.e. remainder) function. Alternatively,m can be computed in this way:

m= x+10, if x≤2, or x−2 otherwise.

•D is the last two digits of the year, but if it is January or February those of the previous year are used.

•Cis for century, and it is the first two digits of year. In our example,C=20.

•From the result f we can obtain the day of the week based on this code:

Day0 = Sunday...Day 6 = Saturday

For example, if=123, thenf%7=4, and thus the day was Thursday. Again, % is the modulus function.

What was the day on June, 1 2005 according to Zeller's formula?

1. There are two career fairs held at a Catholic Chicago University.

1. Fill in the not-hired counts for each fair.

2. Do the samples meet the Success-Failure Conditions?

3. What are the proportions of hired students for each fair?

4. What are the individual standard errors for hired students?

5. What is the difference of proportions of hired students between the two fairs?

6. What is the joint standard error of the two fairs?

7. What is the z-score associated with 90 confidence interval?

8. Compute the confidence interval for this difference of proportions.

1. Which fair should a student attend?

Please solve all parts of the following question. Please show all work and all steps.

1a.) Solve

x' = x + 3y + 2t

y' = x - y + t^2

1b.) Solve

x' + ty = -1

y' + x' = 2

1c.) Solve

x' + y = 3t

y' - tx' = 0

1. Consider the system;

dx/dt=x(1-x/10)-1/2xy

dy/dt=2y(1-y/6)-xy

(a) Find all equilibrium solutions.

(b) Draw the phase plane.

(c) On the phase plane, mark
• all equilibrium solutions.

• the lines where dx/dt =0, dy/dt =0

• an arrow indicating the general direction of the flow of the system in each region on the picture.

(d) If you were given initial conditions x = 3 and y = 4, what do you think would happen to the solution?

Show many ways can you tile a 3xn checkboard with a 2x1 tile? Create a recursive relation and show how to derive  the number of ways.

An eagle is flying horizontally at the altitude of 400 ft with a velocity of 16 ft/s when it drops a dead prey. A frog jumps 2 seconds later at a site facing away from the eagle and 60 ft from the point beneath the original position of the bird. If the frog catches the falling prey at the height of 144 ft, determine the initial speed and angle of inclination of the frog. Please use an advanced math way using vector-valued functions to solve it (Calculus 3).
A note: h = 400 , b = 144 , a = 60

Let y =

U1 =

U2 =

Find the distance from y to the plane in R3 spanned by U1 and U2. Exact answer please.

Engineering system of type k-out-of-n is operational if at least k out of n components are operational. Otherwise, the system fails. Suppose that a k-out-of-n system consists of n identical and independent elements for which the lifetime has Weibull distribution with parameters r and λ. More precisely, if T is a lifetime of a component, P(T ≥ t) = e−λtr, t ≥ 0. Time t is in units of months, and consequently, rate parameter λ is in units (month)−1. Parameter r is dimensionless. Assume that n = 8,k = 4, r = 3/2 and λ = 1/10. (a) Find the probability that a k-out-of-n system is still operational when checked at time t = 3. (b) At the check up at time t = 3 the system was found operational. What is the probability that at that time exactly 5 components were operational? Hint: For each component the probability of the system working at time t is p = e−0.1 t3/2. The probability that a k-out-of-n system is operational corresponds to the tail probability of binomial distribution: IP(X ≥ k), where X is the number of components working. You can do exact binomial calculations or use binocdf in Octave/MATLAB (or dbinom in R, or scipy.stats.binom.cdf in Python when scipy is imported). Be careful with ≤ and <, because of the discrete nature of binomial distribution. Part (b) is straightforward Bayes formula.

Let ? be a ?−algebra in ? and ?:? ⟶[0,∞] a measure on ?. Show that ?(?∪?) = ?(?)+?(?)−?(?∩?) where ?,? ∈?.

Find a particular solution, Yp, of the non-homogenous DE y" + 3y' + 2y = 1/1+e^x