5

11. Exercise 11 on page 565 _ This problem continues the development of ideas for Theorem 1 on page 551. Repeat the procedure Exercise 10 on page 565, but using t...

Question

11. Exercise 11 on page 565 _ This problem continues the development of ideas for Theorem 1 on page 551. Repeat the procedure Exercise 10 on page 565, but using the table below where symbols have replaced the numbers_ The notation in part uses a6t to indicate the number of one-step walks from node 6 to node t, and bt3 for the number of four-step walks from node to node 3. Using the similar notation of a8t to indicate the number of one-step walks from node 8 to node t, and bt2 for the number of

11. Exercise 11 on page 565 _ This problem continues the development of ideas for Theorem 1 on page 551. Repeat the procedure Exercise 10 on page 565, but using the table below where symbols have replaced the numbers_ The notation in part uses a6t to indicate the number of one-step walks from node 6 to node t, and bt3 for the number of four-step walks from node to node 3. Using the similar notation of a8t to indicate the number of one-step walks from node 8 to node t, and bt2 for the number of four-step walks from node to node 2. write a formula for the number of five-step walks from node 8 to node 2_ Generalize to obtain a formula for the number of five-step walks from node to node j Fill in the blanks_ If M is the adjacency matrix for the graph_ the first table in part gives rOw of M and the second table gives column of M". Node Number bj b1 bu b41 0s1 bos bu bxa byj Node Number @ol doz Oad Un5 067 Uot Uta



Answers

In this exercise we will count the number of paths in the $x y$ plane between the origin $(0,0)$ and point $(m, n),$ where $m$ and $n$ are nonnegative integers, such that each path is made up of a series of steps, where each step is a move one unit to the right or a move one unit upward. (No moves to the left or downward are allowed.) Two such paths from $(0,0)$ to $(5,3)$ are illustrated here.
a) Show that each path of the type described can be represented by a bit string consisting of $m$ 0s and $n$ ls, where a 0 represents a move one unit to the right and a 1 represents a move one unit upward.
b) Conclude from part (a) that there are $\left(\begin{array}{c}{m+n} \\ {n}\end{array}\right)$ paths of
the desired type.

Okay, so for this probably and given, uh, are in represents the number of moves is by the frames to her algorithm. Okay, we want to choose into decay to be these smallest integer such that n being the number of desks is less than equal to k times que plus one divided by two cans we want to prove and that are then again, being the number of moves is equal to two times Ah, are and minus K plus two to the K minus one. And additionally, we want to look at our initial conditions so that no disks requires no moves. And that we're having one disc requires one move. Yes, We start this proof off with the bass case. Our base step, we'll take what we have. Zero disk. So if we have our four pegs and no disks on it, we cannot do any moves to solve the puzzle. Right? The puzzles solved just in existence so are not is equal to zero. Okay, And then when we have one disc to get that one does to the fourth peg requires one move right from page one to peg for. So in the case, we have just one desk. It requires one move to solve. So those are the base steps, actually, like just solving initial conditions. Okay, then for recurrence, our recurrence step here we consider what we have n discs. You can't kind of break it down into three broader states. Right? So first, we're going to move the n minus case. Smallest discs. So the smallest discs to the second peg. So if we had our four pegs, it's gonna be some number of larger disks on one. And then some of these smaller ones on peg to left over okay? And that's gonna require are and minus k moves right by the recurrence relation. So next we want to move the K largest disk. So this is everything that's left on paid one to peg for. Okay, we have just three pegs toe work with where you have to leave that second peg alone in order to get the K largest pegs over to the fourth head here. Okay, but Thea Teller of Hanley uses only three pegs who can use that hand. Lie under them to know that it's going to take to the power of K minus one moves Okay. That was the Hanoi algorithm then. Lastly, it's the last thing we want to do here is to move everything that's on that second pegs. We need to move those and minus K. Smallest discs over to peg for. I need to get on top of the rest of them. So to the fourth peg. And again, we're going to do that in the same amount of moves that we move them to the second pig, namely our and minus K moves. Okay, so then we just have to sum up these 12 number of moves, right? Until we do that, we're gonna get that r n. It's equal to all right. In my case, that was moving que smallest pegs to that second smallest this to a second peg, plus the moves it took to move all the discs from the first peg to the fourth pay. That was to the K minus one and plus moving. All the disks from the second peg kay and my case smallest discs to the fourth egg. And we have ah, some common terms here race. We can combine these two terms to get two times are and minus K plus two to the K minus one, and that will complete the proof

We're told that a person can go up as they are one of two ways. They can even take once there at the time. Or they can take to stares at the time. So we're told that if there was one step, you would only take one, there's only one way to do it. Just going up just the one step, right? And if there's two steps that there's two ways you can take one of the time or just take two steps. So what we need to figure out is how many ways are there? If you weigh over three steps? Well if you think about if there's three steps we could take one each time Or we could take one and then two or that we can do to and then one. So there'll be three total ways for three steps. So how about if there's four steps? Well we could take one each time. We could take one the first two times and then take you. Then we could take 12 and then one We could do to one and 1. Or we could take two and two. So as you can see there would be five total weeks. So if you want to start thinking about this kind of as a sequence we have 123 and five. Okay, so now let's take a look at part B. And they said that essentially what we can do in order to find any term of the sequence is we can just add up the previous two terms. So what we want to do is write this as a recursive formula. So in order to find T seven meaning the total number of ways, we just need to add the previous two. So we would do t sub minus two Plus TCN -1. But remember we have to establish the first two terms. Well the first term was one, so when teeth of one is one And the second term was to one teeth of two is equal to two. So now using your calculator, they want you to figure out well how many would there be for the 20 ways? So If you scroll down what you'll find is in your calculator, that piece of 20 would in fact be 10,946. That means if you're over 20 steps this is how many different ways that you can take those 20 steps. Okay, so now important see we if you know this that this sequence is really similar to Fibonacci sequence. The only difference is we're missing that first one. Remember Fibonacci sequence starts with one than one. So the only difference is we're missing the first term because we're missing that first one. Okay so now the last thing party, we want to know, how many different ways were there to be go up um if there were 91 steps. So again, if you go back to your calculator instead of going down to the 20 if you want to go down to the 91st. So in other words, we're looking for T sub 91 And you'll find that it's a very large number would be 7.5 times 10 to the 18th power. There'll be a lot of different ways to climb up those 91 steps.

Okay, so for part A here, we're using triangular numbers which were given to us as que times que plus one over too. Right? So that means is t one is equal to one angle a number 2 to 3. The third number is equal to six. And the 4th 1 is equal to 10. Okay, well, so since tea too, does he cook? Two? Three is less than five, which is the number of disks that were working with which is less than or equal to t three equals six. Then we're going to choose. So we choose K is equal to three eso for notation I'm gonna consider and save pegs A and B. Yes, this is moving disc. See, I from, uh, hang a to B just to simplify how much I'm writing here. So the first move that we're going to do and follows moving the end minus K, which is equal to five minus threes, is gonna be these second smallest disc from the first to the second peg. And so to do that, we'll start by moving our smallest disc to peg three yuan. Our second smallest just get pegged too. So we skipped that one. And then we'll move our second smallest disc to the second peg what we wanted. And they will move our smallest disc on top of that. Okay? And so the next and that's don't want to do is move. The K equals three largest discs rights. We had our and my as K two smallest disk on the second Pagan our doing r K three largest discs onto Hegg four. Okay, so start, I'm moving thes third largest to pig for fourth largest to page three, and I'll move our third largest onto our fourth largest on peg three. Okay, is that frees up our fourth egg? We can move it our largest disk onto hey, four. Nothing in there. We're gonna move our third largest disc from three 21 to the First pig so that we can free up or second largest peg and move that toothy our second largest disc, and move that to the fourth peg. And now we can also move. Our third largest are medium sized disc to the fourth pay as well. So at this point, you would have one of the last pay our three largest and on her second pig are two smallest discs. Okay, so finally, we want to move our smallest two discs to pick for. So last part of the algorithm is to move the N minus. Okay? Equals five minus three is our two smallest discs to peg four. Okay, so we do that first, we're gonna move from Hegg to to pick one our smallest disc that allow us to move from pig to to take four or second smallest disc. And then from one, 24 we just have to move our smallest disc. Okay? And so now everything is stacked up. We can note that we had three moves in this last section seven in this middle and three moves in this first section because it's a total of 13 moves.

We have a gardener who's planting trees in a triangular form, starting with one tree in the first row, two trees in the second row, three in the third row and this continues on for ton rose. So we first want to figure out how many how many trees will be in each row. So again, in the first rollers, one second road to third row three. So this continues. So whatever the row number is, that is the number of trees that are in that row and they spend like I said, it continues on for Ton Rose. We also want to figure out how many total trees that would be so in Ton Rose. We're gonna add up all these numbers, and when we do that, we get a total of 55 trees to write this in summation notation. We're starting with an I value of one, and we're continuing all the way up to 10. And our expression is just simply I


Similar Solved Questions

5 answers
PaioduiaIv1 , aniuses"cUsing a line integral, find the area of the region enclosed by the hyperbola ~? _ ~x2 = 1 and the line y 2_
paioduia Iv1 , aniu ses"c Using a line integral, find the area of the region enclosed by the hyperbola ~? _ ~x2 = 1 and the line y 2_...
5 answers
A motorcycle moves according to the velocity-versus-time graph shown in the figure(Figure 1).Figure1 of 11 1Time t (s)
A motorcycle moves according to the velocity-versus-time graph shown in the figure(Figure 1). Figure 1 of 1 1 1 Time t (s)...
5 answers
There are three parts to this question: would like you to answer the questions regarding three isolated organisms (#1-3) based on the information provided in the table and in the figures: Unknown Source ODsgo with Oxygen ODsgo without Oxygen Fecal matter 0.257 0.255Milk carton0.2430.035Raw chicken0.4130.233List the classification of the organisms represented in the thioglycollate tubes (A-D} this is just the name of the classification; no explanation needed_
There are three parts to this question: would like you to answer the questions regarding three isolated organisms (#1-3) based on the information provided in the table and in the figures: Unknown Source ODsgo with Oxygen ODsgo without Oxygen Fecal matter 0.257 0.255 Milk carton 0.243 0.035 Raw chick...
5 answers
SHQWALLYQUR WQEKFollowing the birth of child parent wants to make an initia# investment Po that will grow to S60000 by the child's 20l birthday. Interest compounded continuously at 5.4%0. What should the intial investment be? Show all your work
SHQWALLYQUR WQEK Following the birth of child parent wants to make an initia# investment Po that will grow to S60000 by the child's 20l birthday. Interest compounded continuously at 5.4%0. What should the intial investment be? Show all your work...
5 answers
Question 42ptsA,B and € are three vectors directed along the x, Y and z axes respectively, as shown If IA| = 4IBl = 3and Icl = 2 then A . (0.4B x €) equals0 7219,2~19,2
Question 4 2pts A,B and € are three vectors directed along the x, Y and z axes respectively, as shown If IA| = 4IBl = 3and Icl = 2 then A . (0.4B x €) equals 0 72 19,2 ~19,2...
1 answers
Figure $2.4 .22$ suggests that the equation $x=-5 \cos x$ has at least three distinct solutions. Use the intermediate value theorem to show that this is true. Then use your calculator to approximate each of these solutions accurate to two decimal places.
Figure $2.4 .22$ suggests that the equation $x=-5 \cos x$ has at least three distinct solutions. Use the intermediate value theorem to show that this is true. Then use your calculator to approximate each of these solutions accurate to two decimal places....
5 answers
Use undetermined coefficients to solve the following initial value problem: (v" + 2v +y = 2e-" y(o) =1 Y(0) = 0Select one: y=(tt-t- 1) exp(-t)y =(t + 1) exp(-t)y=(tt-t-1) exp(t)y =(tt+ 1) exp(-t)y=(tt+t- 1) exp(t)y=(tt+t) exp(-t)y=(tt+t- 1) exp(-t)H. y-(tttttt+ 1) exp(-t)
Use undetermined coefficients to solve the following initial value problem: (v" + 2v +y = 2e-" y(o) =1 Y(0) = 0 Select one: y=(tt-t- 1) exp(-t) y =(t + 1) exp(-t) y=(tt-t-1) exp(t) y =(tt+ 1) exp(-t) y=(tt+t- 1) exp(t) y=(tt+t) exp(-t) y=(tt+t- 1) exp(-t) H. y-(tttttt+ 1) exp(-t)...
5 answers
Use Ihe simplex method to solve the Iinear programming problem. Maximize 2 = 6x1 + 3x2 + X3 subject to 571 " 5x2 X3 $45 +512 3x3 $ 12 X1 20,X2 20,X3 20.Select the correct choice below and, if necessary; fill in the answer boxes to complete your choice.0 AJ The maximum iswhen X1 #Ox-Ox: Ds-L and S2 There is no maximum:
Use Ihe simplex method to solve the Iinear programming problem. Maximize 2 = 6x1 + 3x2 + X3 subject to 571 " 5x2 X3 $45 +512 3x3 $ 12 X1 20,X2 20,X3 20. Select the correct choice below and, if necessary; fill in the answer boxes to complete your choice. 0 AJ The maximum is when X1 #Ox-Ox: Ds-L...
1 answers
In Exercises 1–18, solve each system by the substitution method. $$\left\{\begin{array}{l}{x^{2}+y^{2}=5} \\ {3 x-y=5}\end{array}\right.$$
In Exercises 1–18, solve each system by the substitution method. $$\left\{\begin{array}{l}{x^{2}+y^{2}=5} \\ {3 x-y=5}\end{array}\right.$$...
5 answers
Consider the Poset({1}, {2}, {4}, {1, 2}, {1, 4), (2, 4), (3,4}, {1, 3, 4}, {2, 3, 4), ) Find1. The maximal elements 2. The minimal elements3. All the upper bound of {{2}, {4}} and the least upper bound, if 4. All the lower bounds of {1, 3, 4) and its greatest lower boundexists
Consider the Poset ({1}, {2}, {4}, {1, 2}, {1, 4), (2, 4), (3,4}, {1, 3, 4}, {2, 3, 4), ) Find 1. The maximal elements 2. The minimal elements 3. All the upper bound of {{2}, {4}} and the least upper bound, if 4. All the lower bounds of {1, 3, 4) and its greatest lower bound exists...
4 answers
Continuous Tancom vafabc AhaspoT OT @nc tOrm;ix(674/511) 3 3for 0.24 < % < 132. Calcubic thc stndard dcviation (sigma} of %Yanitiniz:0.1270.7620.9410,2140.4790.9270,6500.2650.0820,.970
continuous Tancom vafabc Ahas poT OT @nc tOrm;ix (674/511) 3 3for 0.24 < % < 132. Calcubic thc stndard dcviation (sigma} of % Yanitiniz: 0.127 0.762 0.941 0,214 0.479 0.927 0,650 0.265 0.082 0,.970...
4 answers
J ~"dr + xdy +xydz where € is r(t) 2costi Zsintj+tk if0<ts21Use Green Theoremn evaluate S[(_ ryldr+xy"dy where € is the boundary of the region bounded by x =J ady=x (5 pts)
J ~"dr + xdy +xydz where € is r(t) 2costi Zsintj+tk if0<ts21 Use Green Theoremn evaluate S[(_ ryldr+xy"dy where € is the boundary of the region bounded by x =J ady=x (5 pts)...
5 answers
LuestionYoutr ansvipartially cortedThe figure shows parallel: plte cadacioct pute 10 7 cM-ang separaticn 2d= The left half of the gap filled with material dielectnc constant dielectric constant Kz 43.3; the bottom the right half filled with materia dielectric constant K3 56.0. What the capacitance?24.1; the top the right half filled with materialMumde4E-L1Units
Luestion Youtr ansvi partially corted The figure shows parallel: plte cadacioct pute 10 7 cM-ang separaticn 2d= The left half of the gap filled with material dielectnc constant dielectric constant Kz 43.3; the bottom the right half filled with materia dielectric constant K3 56.0. What the capacitanc...
5 answers
Solve the following system by the method of reduction_4x16z = 282x - 3y - 32 = 272x + 2y - 32 = -32x + 2y + 2z = - 8Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice_0A X= y= 2 = (Type integers or fractions ) 0 B. X=gy= 2 = (Type integers or fractions:) 0c: There is no solution.
Solve the following system by the method of reduction_ 4x 16z = 28 2x - 3y - 32 = 27 2x + 2y - 32 = -3 2x + 2y + 2z = - 8 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice_ 0A X= y= 2 = (Type integers or fractions ) 0 B. X=gy= 2 = (Type integers or f...
5 answers
In your opinion, do You (hink that ETPUM is an Earthly isolate or &n extraterrestrial isolate" Are therc any microbes, anywhere; that are unable to be cultured? Are there methods t0 study these kinds of microbes?
In your opinion, do You (hink that ETPUM is an Earthly isolate or &n extraterrestrial isolate" Are therc any microbes, anywhere; that are unable to be cultured? Are there methods t0 study these kinds of microbes?...

-- 0.021349--