💡Okay, so I have this hyperbola equation here: (frac{x^2}{64} - frac{y^2}{36} = 1). I need to find the equations of its asymptotes. Hmm, I remember that hyperbolas have asymptotes which are straight lines that the hyperbola approaches but never touches. First, I should recall the standard form of a hyperbola. I think it's either (frac{x^2}{a^2} - frac{y^2}{b^2} = 1) or (frac{y^2}{b^2} - frac{x^2}{a^2} = 1), depending on whether it opens horizontally or vertically. In this case, since the positive term is with (x^2), it must be a horizontally opening hyperbola. So, the standard form is (frac{x^2}{a^2} - frac{y^2}{b^2} = 1).From the given equation, I can identify (a^2 = 64) and (b^2 = 36). To find (a) and (b), I just take the square roots. So, (a = sqrt{64} = 8) and (b = sqrt{36} = 6). Now, I need to find the asymptotes. I remember that for a hyperbola of the form (frac{x^2}{a^2} - frac{y^2}{b^2} = 1), the asymptotes are given by the equations (y = pm frac{b}{a}x). So, plugging in the values of (a) and (b) I found, the slopes would be (frac{6}{8}) and (-frac{6}{8}). Simplifying (frac{6}{8}), I get (frac{3}{4}). So, the slopes are (pm frac{3}{4}).Therefore, the equations of the asymptotes should be (y = frac{3}{4}x) and (y = -frac{3}{4}x). Wait, let me double-check. If I rewrite the hyperbola equation, it's centered at the origin since there are no shifts in the equation. So, the asymptotes should also pass through the origin, which they do in this case. Another way to think about it is that as (x) becomes very large, the hyperbola approaches these lines. So, if I solve for (y) in terms of (x) when the hyperbola equation is set to 1, I can approximate the asymptotes. Let me try that. Starting with (frac{x^2}{64} - frac{y^2}{36} = 1), if I rearrange it, I get (frac{y^2}{36} = frac{x^2}{64} - 1). Then, multiplying both sides by 36, I have (y^2 = 36left(frac{x^2}{64} - 1right)). Taking the square root of both sides, (y = pm sqrt{36left(frac{x^2}{64} - 1right)}). As (x) becomes very large, the "-1" inside the square root becomes negligible compared to (frac{x^2}{64}). So, approximately, (y approx pm sqrt{frac{36x^2}{64}} = pm frac{6}{8}x = pm frac{3}{4}x). That confirms my earlier result.I think I'm confident now that the asymptotes are indeed (y = frac{3}{4}x) and (y = -frac{3}{4}x). So, the equations of the asymptotes are (y = pm frac{3}{4}x).

💡First, I need to understand the given sets ( A ) and ( B ). Set ( A ) is defined as ( {x | x > 3} ), which means it includes all real numbers greater than 3. Set ( B ) is defined as ( {x | 2 < x < 4} ), which includes all real numbers between 2 and 4.Next, I need to find the complement of set ( A ) with respect to the real numbers, denoted as ( complement_R A ). The complement of ( A ) would include all real numbers that are not in ( A ), which means all real numbers less than or equal to 3. Therefore, ( complement_R A = {x | x leq 3} ).Now, I need to find the intersection of ( complement_R A ) and ( B ), which is ( (complement_R A) cap B ). This intersection will include all numbers that are in both ( complement_R A ) and ( B ). Set ( complement_R A ) includes numbers up to 3, and set ( B ) includes numbers greater than 2 and less than 4. The overlap between these two sets is the range of numbers greater than 2 and less than or equal to 3. Therefore, ( (complement_R A) cap B = {x | 2 < x leq 3} ).Finally, I compare this result with the given options:- **A:** ( {x | x leq 3} ) — This is too broad.- **B:** ( {x | 2 < x leq 3} ) — This matches our result.- **C:** ( {x | 3 < x < 4} ) — This is incorrect as it doesn't include numbers up to 3.- **D:** ( {x | x < 4} ) — This is too broad.Thus, the correct answer is **B**.

💡First, I need to understand the problem: I'm looking for a fraction ( x ) such that the ratio of ( x ) to ( frac{2}{6} ) is the same as the ratio of ( frac{3}{4} ) to ( frac{1}{2} ).To set this up mathematically, I can write the proportion:[frac{frac{3}{4}}{frac{1}{2}} = frac{x}{frac{2}{6}}]Next, I'll simplify both sides of the equation. Starting with the left side:[frac{frac{3}{4}}{frac{1}{2}} = frac{3}{4} times frac{2}{1} = frac{6}{4} = frac{3}{2}]Now, the equation looks like this:[frac{3}{2} = frac{x}{frac{2}{6}}]To solve for ( x ), I'll multiply both sides by ( frac{2}{6} ):[x = frac{3}{2} times frac{2}{6} = frac{6}{12} = frac{1}{2}]Therefore, the fraction that has the same ratio to ( frac{2}{6} ) as ( frac{3}{4} ) has to ( frac{1}{2} ) is ( frac{1}{2} ).

💡First, I need to determine the total distance the train must travel to completely cross the bridge. This is the sum of the train's length and the bridge's length.Next, I should convert the train's speed from kilometers per hour to meters per second to ensure the units are consistent for calculating time.Finally, I can calculate the time it takes for the train to cross the bridge by dividing the total distance by the speed in meters per second.

💡Okay, so I'm trying to figure out how many ways Laura can remove exactly 5 blocks from Manya's stack. The stack has 4 layers, and each layer has (4^{k-1}) blocks, where (k) is the layer number from the top. So, the top layer has 1 block, the second has 4, the third has 16, and the bottom has 64 blocks. Each block rests on 4 smaller blocks, each half the size of the larger one. Laura can only remove blocks that have no blocks on top of them, and the order matters.First, I need to understand the structure. The top layer has just 1 block. When Laura removes that block, it should expose the 4 blocks in the second layer. Then, removing any of those 4 blocks would expose 4 more blocks in the third layer, and so on. So, each time a block is removed, it potentially exposes 4 new blocks. But wait, if Laura removes a block from a lower layer, does it expose more blocks or not? Hmm, actually, if you remove a block from a lower layer, it doesn't expose any new blocks because the blocks above it are already exposed. So, the number of exposed blocks depends on how Laura removes the blocks.Let me think about this step by step. Initially, only the top block is exposed. So, Laura has only 1 choice for the first removal. After removing the top block, the 4 blocks in the second layer become exposed. So, for the second removal, Laura has 4 choices. Now, if Laura removes one of those 4 blocks, that would expose 4 more blocks in the third layer. So, for the third removal, Laura would have 4 (remaining from the second layer) + 4 (exposed from the third layer) = 8 choices? Wait, no, because once you remove a block from the second layer, only that specific block's 4 children are exposed, not all 4 blocks from the third layer. So, actually, after removing the first block, you have 4 exposed. After removing one of those, you have 3 remaining in the second layer and 4 exposed in the third layer, so total 7 exposed blocks. So, for the third removal, Laura has 7 choices.Similarly, after the third removal, if Laura removes a block from the second layer, she'll expose another 4 blocks in the third layer, making the total exposed blocks 10. If she removes a block from the third layer, she'll expose 4 blocks in the fourth layer, but wait, the fourth layer is the bottom layer, so removing a block from the third layer would expose 4 blocks in the fourth layer. But Laura can only remove blocks that have no blocks on top of them, so if she removes a block from the third layer, the blocks in the fourth layer are now exposed.Wait, this is getting a bit complicated. Maybe I should model this as a tree. The stack is like a quaternary tree with 4 layers. Each node has 4 children. Laura is essentially performing a traversal where she can only remove leaves (nodes with no children). Each time she removes a leaf, its parent becomes a leaf if it wasn't already. So, the number of leaves increases as she removes blocks.But in this case, the tree is inverted because the top block is the root, and each subsequent layer is a level in the tree. So, the root has 4 children, each of those has 4 children, and so on. So, initially, only the root is a leaf (since it's the top block). When Laura removes the root, its 4 children become leaves. Then, each time she removes a leaf, its 4 children become leaves, but only if they weren't already.Wait, but in this case, the bottom layer is the fourth layer, so if Laura removes a block from the third layer, its 4 children in the fourth layer become leaves. But the fourth layer is the bottom, so those blocks don't have any children. So, removing a block from the third layer would expose 4 new leaves in the fourth layer.But Laura can also choose to remove blocks from the second or third layers, which would expose more leaves. So, the number of leaves increases as she removes blocks.So, starting with 1 leaf (the top block). After removing it, 4 leaves are exposed. So, for the first removal, 1 choice. For the second removal, 4 choices. For the third removal, if Laura removes one of the second layer blocks, that would expose 4 more blocks in the third layer, making total leaves 4 - 1 + 4 = 7. So, 7 choices for the third removal.Similarly, for the fourth removal, if Laura removes a second layer block, it would expose another 4 third layer blocks, making total leaves 7 - 1 + 4 = 10. If she removes a third layer block, it would expose 4 fourth layer blocks, making total leaves 7 - 1 + 4 = 10 as well. So, regardless of which block she removes, the number of leaves increases by 3 each time.Wait, so the number of leaves after n removals is 1 + 3(n - 1). So, for n=1, 1 leaf. For n=2, 4 leaves. For n=3, 7 leaves. For n=4, 10 leaves. For n=5, 13 leaves. So, the number of ways to remove 5 blocks would be the product of the number of choices at each step: 1 * 4 * 7 * 10 * 13.Calculating that: 1 * 4 = 4, 4 * 7 = 28, 28 * 10 = 280, 280 * 13 = 3640.But wait, is that correct? Because when Laura removes a block from the bottom layer, does it not expose any new blocks? Or does it just stay as is?Wait, the bottom layer is the fourth layer, which has 64 blocks. Each of these blocks doesn't have any children because it's the bottom. So, if Laura removes a block from the third layer, it exposes 4 blocks in the fourth layer. But if she removes a block from the fourth layer, it doesn't expose any new blocks because there are no blocks below it.So, in the earlier reasoning, when we assumed that each removal increases the number of leaves by 3, that's only true if the removed block is not from the bottom layer. If Laura removes a block from the bottom layer, it doesn't expose any new blocks, so the number of leaves doesn't increase.Therefore, the initial calculation of 1 * 4 * 7 * 10 * 13 = 3640 might be overcounting because it assumes that each removal increases the number of leaves by 3, but in reality, if Laura removes a block from the bottom layer, the number of leaves doesn't increase.So, we need to adjust for that. How?Well, let's think about it. The total number of ways without considering the bottom layer is 3640. But some of these ways involve removing blocks from the bottom layer before all the blocks above them have been removed. So, we need to subtract those cases where Laura removes a block from the bottom layer before it's supposed to be exposed.Wait, but how do we calculate that? It might be complicated because Laura could remove a bottom layer block at any step after it's exposed. So, perhaps we need to use inclusion-exclusion or something.Alternatively, maybe we can model this as a generating tree where each node branches into 4 children, and we're counting the number of sequences of 5 nodes where each node is a leaf at the time of removal.But that might be too abstract. Maybe another approach is to realize that the number of ways to remove 5 blocks is equal to the number of linear extensions of the poset defined by the stack, truncated at 5 elements.But that might be too advanced for me right now. Let me try a different approach.Let me think about the possible sequences of removals. Each removal can be from the top layer, second layer, third layer, or fourth layer, but with the constraint that you can't remove a block from a lower layer until all the blocks above it have been removed.Wait, no, actually, you can remove a block from a lower layer as long as it's a leaf, meaning that all the blocks above it have been removed. So, for example, to remove a block from the second layer, you need to have removed the block above it in the first layer. Similarly, to remove a block from the third layer, you need to have removed the block above it in the second layer, and so on.So, the removal sequence must respect the tree structure, meaning that you can't remove a child before its parent.But in this case, the tree is inverted, so the root is at the top, and the children are below. So, the parent is above the child, so you can't remove a child before its parent. So, the removal sequence must be such that you remove the parent before any of its children.Therefore, the number of ways to remove 5 blocks is equal to the number of sequences of 5 nodes in the tree where each node is removed after its parent.This is similar to counting the number of chains in the tree of length 5, but considering the order of removals.Wait, no, it's more like counting the number of linear extensions of the tree poset, but only for the first 5 elements.But I'm not sure about that. Maybe another way is to realize that each removal corresponds to choosing a leaf in the current tree and removing it, which then may expose new leaves.So, starting with 1 leaf (the root). After removing the root, we have 4 leaves (the second layer). After removing one of those, we have 3 leaves in the second layer and 4 in the third layer, totaling 7. After removing another, it could be from the second or third layer, leading to 10 leaves, and so on.But as I thought earlier, the number of leaves after n removals is 1 + 3(n - 1). So, for n=5, it's 13 leaves. So, the total number of ways is 1 * 4 * 7 * 10 * 13 = 3640.But this assumes that every removal increases the number of leaves by 3, which is only true if the removed block is not from the bottom layer. If Laura removes a block from the bottom layer, it doesn't expose any new blocks, so the number of leaves doesn't increase.Therefore, the initial calculation overcounts the number of ways because it assumes that each removal increases the number of leaves by 3, but in reality, some removals (from the bottom layer) don't increase the number of leaves.So, how do we adjust for that?Well, let's think about how many sequences involve removing a block from the bottom layer before all the blocks above it have been removed. Each such sequence would have fewer leaves exposed than the initial calculation assumes.But it's tricky to calculate exactly how many such sequences there are because Laura could remove a bottom layer block at any point after it's exposed, which could be as early as the third removal.Wait, actually, the bottom layer blocks are only exposed after their parent in the third layer has been removed. So, the earliest Laura can remove a bottom layer block is after the third removal.So, in the sequence of 5 removals, Laura could remove a bottom layer block on the third, fourth, or fifth removal.Each time she removes a bottom layer block, it doesn't expose any new blocks, so the number of leaves doesn't increase by 3, but only by 0.Therefore, the number of leaves after each removal depends on whether the removed block was from the bottom layer or not.This complicates the calculation because the number of leaves isn't a simple arithmetic progression anymore.Maybe a better approach is to model this as a generating function or use recursion.Let me define ( f(n, k) ) as the number of ways to remove ( n ) blocks with ( k ) exposed blocks. Then, we can build a recurrence relation.Initially, ( f(0, 1) = 1 ) because with 0 removals, there's 1 exposed block.For each step, if we have ( k ) exposed blocks, we can remove any of them, and if the removed block is not from the bottom layer, it will expose 4 new blocks, increasing the number of exposed blocks by 3 (since we removed 1 and exposed 4). If the removed block is from the bottom layer, it doesn't expose any new blocks, so the number of exposed blocks remains ( k - 1 ).But wait, actually, when you remove a block, regardless of its layer, you remove 1 exposed block. If it's not from the bottom layer, you add 4 new exposed blocks, so the total becomes ( k - 1 + 4 = k + 3 ). If it is from the bottom layer, you just remove 1, so the total becomes ( k - 1 ).Therefore, the recurrence relation is:( f(n, k) = ) (number of ways to remove a non-bottom block) * ( f(n - 1, k - 3) ) + (number of ways to remove a bottom block) * ( f(n - 1, k + 1) )Wait, no, actually, it's the other way around. If you have ( k ) exposed blocks, and you remove one, if it's a non-bottom block, you add 4 new exposed blocks, so the next state is ( k + 3 ). If it's a bottom block, you just remove 1, so the next state is ( k - 1 ).But actually, the number of exposed blocks after removal depends on whether the removed block was a bottom block or not. So, to model this, we need to know how many of the ( k ) exposed blocks are bottom blocks.Wait, this is getting too complicated. Maybe another approach is needed.Alternatively, perhaps we can think of the problem as a labeled tree and count the number of ways to perform a traversal where we remove nodes one by one, always removing a leaf. This is similar to a Prüfer sequence, but I'm not sure.Wait, another idea: the number of ways to remove 5 blocks is equal to the number of chains of length 5 in the tree, where each chain is a sequence of nodes where each node is a child of the previous one. But no, that's not exactly it because Laura can remove any exposed block, not necessarily following a chain.Wait, maybe it's better to think in terms of permutations. Each removal sequence is a permutation of 5 blocks where each block is removed after its parent.But I'm not sure how to count that.Wait, perhaps I can use the concept of hook lengths in trees. In a rooted tree, the number of linear extensions is given by the hook length formula. But I'm not sure if that applies here.Alternatively, maybe I can use exponential generating functions. The generating function for the number of ways to remove blocks would be the product over all nodes of (1 + x + x^2 + ...), but I'm not sure.Wait, maybe I can think of each block as contributing a factor to the generating function based on when it can be removed. The root can be removed at any time, but its children can only be removed after the root is removed, and so on.But this might be too involved.Wait, going back to the initial idea, if we assume that each removal increases the number of leaves by 3, except when removing a bottom block, which doesn't increase it. So, the total number of ways is 1 * 4 * 7 * 10 * 13 = 3640, but we need to subtract the cases where Laura removes a bottom block before all the blocks above it have been removed.But how many such cases are there?Well, the bottom layer has 64 blocks, each of which is a child of a block in the third layer. So, to remove a bottom block, Laura must have already removed its parent in the third layer.But in the sequence of 5 removals, Laura could remove a bottom block as early as the third removal. So, let's consider the number of sequences where Laura removes a bottom block on the third, fourth, or fifth removal.For each such case, the number of leaves doesn't increase by 3, but only by 0.So, let's calculate the total number of sequences without considering the bottom layer: 1 * 4 * 7 * 10 * 13 = 3640.Now, let's calculate the number of sequences where Laura removes a bottom block at some point. Let's denote this as S.To calculate S, we can consider the number of ways Laura can remove a bottom block on the third, fourth, or fifth removal.Case 1: Laura removes a bottom block on the third removal.To do this, she must have removed the parent block in the third layer on the second or third removal.Wait, no, to remove a bottom block on the third removal, she must have removed its parent in the third layer on the second removal.Because the parent in the third layer must be removed before the bottom block can be exposed.So, the sequence would be:1. Remove the root (1 way).2. Remove a second layer block (4 ways).3. Remove a third layer block (4 ways, since each second layer block has 4 children).4. Now, the bottom layer block is exposed. So, for the third removal, Laura could remove the third layer block, exposing the bottom layer block, and then on the fourth removal, remove the bottom layer block.Wait, no, if Laura removes the third layer block on the third removal, then the bottom layer block becomes exposed, and she can remove it on the fourth removal.But in this case, the third removal is a third layer block, and the fourth removal is a bottom layer block.Similarly, Laura could remove the third layer block on the second removal, exposing the bottom layer block, and then remove it on the third removal.Wait, no, because the third layer block is a child of the second layer block. So, to remove a third layer block, Laura must have already removed its parent in the second layer.So, the sequence would be:1. Remove root (1 way).2. Remove a second layer block (4 ways).3. Remove a third layer block (4 ways).4. Remove a bottom layer block (4 ways).But this is a sequence of 4 removals, but we need sequences of 5 removals. So, Laura could continue removing more blocks after that.Wait, maybe I'm overcomplicating.Alternatively, perhaps the number of sequences where Laura removes a bottom block on the third removal is:1. Remove root (1 way).2. Remove a second layer block (4 ways).3. Remove a third layer block (4 ways).4. Remove a bottom layer block (4 ways).But this is only 4 removals. To make it 5, Laura would have to remove another block, which could be another bottom layer block or another third layer block.Wait, but if Laura removes a bottom layer block on the fourth removal, then on the fifth removal, she could remove another bottom layer block or another third layer block.But this is getting too involved.Maybe a better approach is to realize that the number of sequences where Laura removes a bottom block is equal to the number of ways to choose when to remove the bottom block, multiplied by the number of ways to remove the other blocks.But I'm not sure.Alternatively, maybe the initial calculation of 3640 is correct because even though removing a bottom block doesn't expose new blocks, the number of leaves is still increasing because other blocks are being removed and exposing new ones.Wait, but if Laura removes a bottom block, it doesn't expose new blocks, so the number of leaves decreases by 1 instead of increasing by 3.Therefore, the total number of leaves after each removal is not 1 + 3(n - 1), but depends on how many bottom blocks have been removed.So, the initial assumption was incorrect because it didn't account for the fact that removing a bottom block doesn't increase the number of leaves.Therefore, the correct number of ways is less than 3640.But how much less?Maybe we can calculate it by considering that each time Laura removes a bottom block, the number of leaves decreases by 1 instead of increasing by 3. So, the total number of leaves after n removals is 1 + 3(n - 1) - 2m, where m is the number of bottom blocks removed.But since we're only removing 5 blocks, m can be 0, 1, 2, 3, 4, or 5.But this seems too vague.Alternatively, perhaps we can model this as a state machine where each state is the number of exposed blocks, and transitions depend on whether Laura removes a bottom block or not.But this might be too time-consuming.Wait, maybe I can use the principle of inclusion-exclusion. The total number of sequences without any restrictions is 3640. Now, we need to subtract the number of sequences where Laura removes a bottom block before it's supposed to be exposed.But how?Alternatively, perhaps the number of valid sequences is equal to the number of sequences where Laura never removes a bottom block until all the blocks above it have been removed.But that's essentially the same as the initial calculation, which assumes that each removal increases the number of leaves by 3.Wait, but in reality, Laura can remove a bottom block at any time after its parent has been removed, which could be before all the blocks above it have been removed.So, the initial calculation overcounts because it assumes that each removal increases the number of leaves by 3, but in reality, some removals (from the bottom layer) don't increase the number of leaves.Therefore, the correct number of ways is less than 3640.But how much less?Maybe we can calculate it by considering the number of sequences where Laura removes a bottom block at each possible step and subtracting those.But this might be complicated.Alternatively, perhaps the correct number is 3640 minus the number of sequences where Laura removes a bottom block before all the blocks above it have been removed.But I'm not sure how to calculate that.Wait, maybe another approach is to realize that the number of ways to remove 5 blocks is equal to the number of ways to choose 5 blocks such that each block is removed after its parent, multiplied by the number of permutations of these 5 blocks.But that might not be correct because the order matters, and the constraints are on the order.Wait, perhaps it's better to think of this as a tree and count the number of chains of length 5, but considering the order of removals.But I'm not sure.Wait, maybe I can use the concept of hook lengths. In a rooted tree, the number of linear extensions is given by the product over all nodes of (size of the subtree rooted at the node). But I'm not sure.Alternatively, maybe I can use the fact that the number of ways to remove the blocks is equal to the number of ways to perform a breadth-first search or something similar, but I don't think that's the case.Wait, another idea: the number of ways to remove 5 blocks is equal to the number of ways to choose 5 nodes in the tree such that each node is removed after its parent, and then multiply by the number of linear extensions for those 5 nodes.But I'm not sure.Wait, maybe I can use recursion. Let's define ( f(n) ) as the number of ways to remove n blocks. Then, ( f(1) = 1 ), ( f(2) = 4 ), ( f(3) = 4 * 7 = 28 ), ( f(4) = 28 * 10 = 280 ), ( f(5) = 280 * 13 = 3640 ). But this assumes that each removal increases the number of leaves by 3, which is only true if Laura doesn't remove any bottom blocks.But since Laura can remove bottom blocks, which don't increase the number of leaves, the actual number of ways is less than 3640.Wait, but how much less?Maybe we can calculate the number of sequences where Laura removes at least one bottom block and subtract that from 3640.But to calculate that, we need to know how many sequences involve removing a bottom block.But this seems too involved.Alternatively, maybe the correct answer is 3640, and the initial concern about overcounting is unfounded because the model assumes that each removal increases the number of leaves by 3, which is only true if Laura doesn't remove any bottom blocks. But in reality, Laura can remove bottom blocks, which don't increase the number of leaves, so the number of leaves is actually less than 1 + 3(n - 1).Therefore, the initial calculation overcounts, and the correct number is less than 3640.But without a precise way to calculate the exact number, I'm stuck.Wait, maybe I can look for a pattern or a formula.In similar problems, the number of ways to remove blocks from a stack where each removal can expose new blocks is given by the product of the number of exposed blocks at each step.In this case, the number of exposed blocks after each removal is 1, 4, 7, 10, 13, etc., increasing by 3 each time, assuming that each removal is from a non-bottom layer.But if Laura removes a bottom layer block, the number of exposed blocks doesn't increase.Therefore, the total number of ways is the sum over all possible numbers of bottom blocks removed, multiplied by the number of ways to choose when to remove them.But this seems too vague.Alternatively, perhaps the correct answer is 3640, and the overcounting is negligible or accounted for in the model.Wait, maybe I should accept that the initial calculation is correct because it models the number of exposed blocks as increasing by 3 each time, which is an upper bound, and the actual number is less, but without a better method, 3640 might be the intended answer.But I'm not sure.Wait, another idea: the number of ways to remove 5 blocks is equal to the number of ways to choose a sequence of 5 blocks where each block is removed after its parent. This is equivalent to the number of chains of length 5 in the tree, considering the order.But in a quaternary tree of height 4, the number of such chains is ( 4^4 = 256 ), but that's only for chains from the root to a leaf. But Laura can remove any blocks, not necessarily forming a chain.Wait, no, because Laura can remove any exposed block, which could be from different branches.Wait, maybe the number of ways is equal to the number of permutations of 5 blocks where each block is removed after its parent. This is similar to the number of linear extensions of the tree poset, but only for 5 elements.But I don't know how to calculate that.Alternatively, maybe the number of ways is equal to the number of ways to choose 5 blocks such that they form a forest where each tree is a chain, and then multiply by the number of permutations.But I'm not sure.Wait, maybe I can use the fact that the number of ways to remove n blocks from a tree is equal to the product of the number of choices at each step, which is 1 * 4 * 7 * 10 * 13 = 3640, assuming that each removal increases the number of leaves by 3.But since Laura can remove bottom blocks, which don't increase the number of leaves, the actual number is less.But without a precise way to calculate it, I might have to go with 3640 as the answer, acknowledging that it's an upper bound.But I'm not sure. Maybe the correct answer is 3640.Wait, but in the initial problem, it's specified that Laura removes blocks one at a time, removing only blocks that currently have no blocks on top of them. So, the order matters, and the number of ways is the product of the number of choices at each step.So, starting with 1 choice, then 4, then 7, then 10, then 13, giving 1 * 4 * 7 * 10 * 13 = 3640.But this assumes that each removal increases the number of leaves by 3, which is only true if Laura doesn't remove any bottom blocks.But Laura can remove bottom blocks, which don't increase the number of leaves, so the number of leaves after each removal is less than 1 + 3(n - 1).Therefore, the initial calculation overcounts.But how much?Wait, maybe the number of ways is 3640 minus the number of sequences where Laura removes a bottom block.But I don't know how to calculate that.Alternatively, maybe the correct answer is 3640, and the overcounting is acceptable because the problem doesn't specify that Laura can't remove bottom blocks early.Wait, but the problem says Laura removes blocks one at a time, removing only blocks that currently have no blocks on top of them. So, she can remove a bottom block as soon as its parent has been removed, which could be as early as the third removal.Therefore, the number of leaves after each removal is not necessarily 1 + 3(n - 1), but can be less if Laura removes a bottom block.Therefore, the initial calculation of 3640 is an overcount.But without a precise way to calculate the exact number, I might have to conclude that the answer is 3640.But I'm not sure. Maybe the correct answer is 3640.Wait, but in the initial problem, the stack has 4 layers, and Laura is removing 5 blocks. The bottom layer has 64 blocks, so it's possible that Laura could remove multiple bottom blocks in the 5 removals.But each time she removes a bottom block, it doesn't expose any new blocks, so the number of leaves decreases by 1.Therefore, the number of leaves after each removal depends on how many bottom blocks have been removed.But this is too complicated to model without a more advanced approach.Given the time I've spent on this, I think the answer is 3640, but I'm not entirely confident because of the overcounting issue.

💡First, I need to determine the total cost Sahil incurred for purchasing and preparing the machine for sale. This includes the initial purchase price, repair costs, and transportation charges.Next, I'll calculate the total cost by adding these amounts together.After obtaining the total cost, I'll calculate the desired profit, which is 50% of the total cost.Finally, I'll add the profit to the total cost to find the selling price at which Sahil sold the machine.