10. The Tower of Hanoi is a puzzle consisting of a board with three dowels and a collection of n disks of n different radii. The disks have holes drilled through their centers so they can fit on the dowels on the board. Initially, all the disks are on the first dowel arranged in order of their sizes, with the largest one being at the bottom, and the smallest one on the top. The object is to move all the disks to another dowel in as few moves as possible. Each move consists of taking the top disk from one of the stacks and placing it on another with the added condition that you may not place a larger disk on top of a smaller one. Prove: For every n ≥ 1, the Tower of Hanoi puzzle with n disks can be solved in 2^n − 1 moves.

1.Suppose n and k are two positive integers. Pick a uniformly random lattice path from (0, 0) to (n, k). What is the probability that the first step is ‘up’?

Let L be the set of all languages over alphabet {0}. Show that L is uncountable, using a proof by diagonalization.

1. Explain/Define the classical experiment in terms of the following (provide references):     

1. Independent and dependent variables     

2. Pre-testing and Post-testing     

3. Experimental and Control groups

1. Let G be a k-regular bipartite graph. Use Corollary 3.1.13 to prove that G can be decomposed into r-factors iff r divides k.

Suppose a sample of 49 paired differences that have been randomly selected from a normally distributed population of paired differences yields a sample mean d¯ =4.6d¯ =4.6 of and a sample standard deviation of sd = 7.6.

(a) Calculate a 95 percent confidence interval for µd = µ1 – µ2. Can we be 95 percent confident that the difference between µ1 and µ2 is greater than 0? (Round your answers to 2 decimal places.)

Confidence interval = [ ,  ] ; (Click to select)YesNo

(b) Test the null hypothesis H0: µd = 0 versus the alternative hypothesis Ha: µd ≠ 0 by setting α equal to .10, .05, .01, and .001. How much evidence is there that µd differs from 0? What does this say about how µ1 and µ2 compare? (Round your answer to 3 decimal places.)

(c) The p-value for testing H0: µd < 3 versus Ha: µd > 3 equals .0735. Use the p-value to test these hypotheses with α equal to .10, .05, .01, and .001. How much evidence is there that µd exceeds 3? What does this say about the size of the difference between µ1 and µ2? (Round your answer to 3 decimal places.)

rev: 07_14_2017_QC_CS-93578, 12_08_2018_QC_CS-150993

1.six men and six women sit around a desk, what's the probability that A sits next to B?

A.1/6 B.1/12 C.1/11 D.2/11

2.In the same setting. Given A is a woman and B is a man, What's the probability that A sits next to B?

A.2/11 B.1/6 C.1/12 D.1/11

construct a bijective function of f:[0, inf) -> R   

Let S = {a, b, c}. Draw a graph whose vertex set is P(S) and for which the subsets A and B of S are adjacent if and only if A ⊂ B and |A| = |B| − 1.

(a) How many vertices and edges does this graph have?

(b) Can you name this graph?

(c) Is this graph connected?

(d) Does it have a perfect matching? If yes, draw a sketch of the matching.

(e) Does it have a Hamiltonian cycle? If yes, draw a sketch of the cycle.

y' = 2 + t^2 + y^2 0<t<1 y(0)=0
use the euler method to determine step size (h) to keep global truncation error below .0001

Solve: x²y’’ - 7xy’ + 15y = 4x⁶

Using the function f(x)=ln(1+x)

a. Find the 8 degree taylor polynomial centered at 0 and simplify.

b. using your 8th degree taylor polynomial and taylors inequality, find the magnitude of the maximum possible error on [0,0.1]

c.approximate ln(1.1) using your 8th degree taylor polynomial. what is the actual error? is it smaller than your estimated error?Round answer to enough decimal places so you can determine.

d. create a plot of the function f(x)=ln(1+x) along with your taylor polynomial. Based on the plot what appears to be the interval of convergence? explain.

Find sequences that satisfy the following or explain why no such sequence exists:

a) A sequence with subsequences converging to 1, 2, and 3.

b) A sequence that is bounded above, but has no convergent subsequence.

c) A sequence that has a convergent subsequence but is unbounded (note: unbounded
means not bounded below or not bounded above.

d) A sequence that is monotonic and bounded, but does not converge.