💡Okay, so I have this problem about assigning roles in a play. Let me try to understand it step by step. The play has three different male roles, three different female roles, and three different roles that can be played by either gender. The rules are that only men can be assigned to male roles, and only women can be assigned to female roles. We have seven men and eight women auditioning. I need to figure out in how many ways the nine roles can be assigned.First, I think I should break down the problem into parts. There are three types of roles: male-only, female-only, and gender-neutral. So, I need to assign the male roles first, then the female roles, and finally the gender-neutral roles. Starting with the male roles: there are three distinct male roles and seven men auditioning. Since each role is different, the order in which we assign the men matters. So, for the first male role, I can choose any of the seven men. Once I've assigned someone to the first role, there are six men left for the second role. Then, for the third role, there are five men remaining. So, the number of ways to assign the male roles should be 7 multiplied by 6 multiplied by 5. Let me write that down: 7 × 6 × 5. Calculating that gives me 210 ways to assign the male roles.Next, moving on to the female roles. There are three distinct female roles and eight women auditioning. Similar to the male roles, the order matters here too. For the first female role, I can choose any of the eight women. After assigning someone to the first role, there are seven women left for the second role, and then six women remaining for the third role. So, the number of ways to assign the female roles should be 8 × 7 × 6. Let me compute that: 8 × 7 is 56, and 56 × 6 is 336. So, there are 336 ways to assign the female roles.Now, the tricky part is the gender-neutral roles. There are three of these roles, and they can be assigned to either men or women. However, I need to remember that some men and women have already been assigned to their respective roles. Specifically, we've already assigned three men to male roles and three women to female roles. So, how many men and women are left for the gender-neutral roles?Originally, there were seven men and eight women. After assigning three men and three women, the remaining number of men is 7 - 3 = 4, and the remaining number of women is 8 - 3 = 5. So, there are 4 men and 5 women left, making a total of 4 + 5 = 9 auditioners available for the three gender-neutral roles.Since these roles can be played by either gender, any of the remaining 9 auditioners can be assigned to these roles. Again, since the roles are distinct, the order matters. So, for the first gender-neutral role, I can choose any of the 9 remaining auditioners. After assigning someone to the first role, there are 8 left for the second role, and then 7 left for the third role. Therefore, the number of ways to assign the gender-neutral roles should be 9 × 8 × 7. Let me calculate that: 9 × 8 is 72, and 72 × 7 is 504. So, there are 504 ways to assign the gender-neutral roles.Now, to find the total number of ways to assign all nine roles, I need to multiply the number of ways to assign each type of role together. That is, the number of ways to assign male roles multiplied by the number of ways to assign female roles multiplied by the number of ways to assign gender-neutral roles. So, that would be 210 × 336 × 504.Let me compute that step by step. First, 210 multiplied by 336. Let's see, 210 × 300 is 63,000, and 210 × 36 is 7,560. Adding those together gives 63,000 + 7,560 = 70,560. So, 210 × 336 = 70,560.Now, I need to multiply that result by 504. So, 70,560 × 504. Hmm, that's a bit more complex. Let me break it down. 70,560 × 500 is 35,280,000, and 70,560 × 4 is 282,240. Adding those together gives 35,280,000 + 282,240 = 35,562,240.So, putting it all together, the total number of ways to assign all nine roles is 35,562,240.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting with the male roles: 7 × 6 × 5 is indeed 210. Female roles: 8 × 7 × 6 is 336. Gender-neutral roles: 9 × 8 × 7 is 504. Multiplying those together: 210 × 336 is 70,560, and 70,560 × 504 is 35,562,240. Hmm, that seems correct. I don't see any errors in my reasoning or calculations. So, I think the final answer is 35,562,240 ways to assign the roles.

💡Alright, so I came across this interesting problem about an election in Ivanland. Let me try to wrap my head around it. The problem states:"Let ( k ) be a fixed integer. In the town of Ivanland, there are at least ( k+1 ) citizens standing on a plane such that the distances between any two citizens are distinct. An election is to be held such that every citizen votes the ( k )-th closest citizen to be the president. What is the maximal number of votes a citizen can have?"Hmm, okay. So, we have citizens placed on a plane with all pairwise distances being unique. Each citizen votes for their ( k )-th closest neighbor. We need to find the maximum number of votes any single citizen can receive.First, let me make sure I understand the setup correctly. Each citizen has a unique distance to every other citizen, so there are no ties in the distances. When each citizen votes, they look at the distances to all other citizens, sort them from closest to farthest, and then vote for the one who is exactly the ( k )-th closest. So, if ( k = 1 ), everyone votes for their closest neighbor. If ( k = 2 ), everyone votes for their second closest neighbor, and so on.The question is asking for the maximum number of votes a single citizen can receive. So, we need to find a configuration of citizens on the plane where one citizen is the ( k )-th closest neighbor to as many other citizens as possible.Let me think about how to approach this. Maybe I can start with small values of ( k ) and see if I can find a pattern or a strategy that can be generalized.**Case 1: ( k = 1 )**If ( k = 1 ), each citizen votes for their closest neighbor. So, the question becomes: what's the maximum number of people who can have the same closest neighbor? In other words, how many people can have the same person as their closest neighbor?I remember that in graph theory, this is related to the concept of a "dominating set" or something like that. But in geometry, it's about how many points can have another point as their nearest neighbor.I think the maximum number is 5. Wait, why 5? Because in a plane, you can arrange points around a central point such that each surrounding point is closer to the center than to any other surrounding point. If you have 6 points around the center, each at the same distance, then each surrounding point would be equidistant to two others, but the problem states that all distances are distinct. So, maybe with 5 points, you can arrange them such that each is closer to the center than to any other surrounding point.Wait, but the problem says "at least ( k+1 )" citizens, which for ( k=1 ) is at least 2. But we need the maximum number of votes, so maybe arranging more points.But actually, for ( k=1 ), the maximum number of votes a citizen can get is 5. Because if you have a central point and 5 points around it, each of the 5 points can have the central point as their closest neighbor. But if you try to add a 6th point, it might end up being closer to one of the existing surrounding points than to the center. So, 5 is the maximum.But wait, the problem says "at least ( k+1 )" citizens. For ( k=1 ), that's at least 2. But we can have more, right? So, in this case, the maximum number of votes is 5.**Case 2: ( k = 2 )**Now, for ( k=2 ), each citizen votes for their second closest neighbor. So, we need to find how many people can have the same person as their second closest neighbor.This seems trickier. Maybe we can use a similar approach. Place a central point and arrange other points around it. But now, each surrounding point should have the central point as their second closest neighbor, not their first.So, how can we arrange points such that each surrounding point's closest neighbor is someone else, and the second closest is the center.Maybe arrange points in two layers: an inner layer and an outer layer. The inner layer points are closer to the center, and the outer layer points are farther away. Then, for each outer layer point, their closest neighbor is another outer layer point, and their second closest is the center.But wait, if the outer layer points are all equidistant from the center, but we need all pairwise distances to be distinct. So, we can't have them equidistant. Maybe arrange them in a circle with slightly varying radii so that each outer point is closer to the center than to any other outer point.Wait, no. If they are in a circle, each outer point is equidistant to the center, but if we vary the radii, some might be closer to the center than others, but we have to ensure that for each outer point, their closest neighbor is another outer point, not the center.This is getting complicated. Maybe another approach.Alternatively, think about placing multiple points around the central point such that for each surrounding point, the central point is their second closest. So, each surrounding point has another surrounding point as their closest neighbor, and the central point as their second closest.To achieve this, maybe arrange the surrounding points in pairs, each pair close to each other, but each pair is far enough from the center that the distance to the center is greater than the distance within the pair.But then, how many such pairs can we have? If we have too many pairs, the distance between pairs might become smaller than the distance to the center, which would mess up the ordering.Alternatively, arrange the surrounding points in a regular polygon around the center, but with a radius such that the distance between adjacent surrounding points is less than the distance from the surrounding points to the center.Wait, in a regular polygon, the distance between adjacent points is ( 2r sin(pi/n) ), where ( r ) is the radius and ( n ) is the number of sides. The distance from a surrounding point to the center is ( r ). So, to have the distance between surrounding points less than ( r ), we need ( 2 sin(pi/n) < 1 ), which implies ( sin(pi/n) < 1/2 ). So, ( pi/n < pi/6 ), which implies ( n > 6 ).Wait, that can't be. If ( n > 6 ), then the distance between adjacent points is less than ( r ). So, for ( n = 7 ), the distance between adjacent points is less than ( r ), meaning each surrounding point's closest neighbor is another surrounding point, and the second closest is the center.But we need all distances to be distinct. So, if we have 7 surrounding points, each with slightly different radii, but arranged such that the distance between any two surrounding points is less than their distance to the center.But this might not be straightforward because varying the radii would affect the distances between surrounding points.Alternatively, maybe arrange the surrounding points not in a regular polygon but in a way that each has a unique distance to the center and unique distances to other surrounding points.This is getting too vague. Maybe I should look for a pattern or think about how the number of votes scales with ( k ).Wait, in the first case, ( k=1 ), the maximum number of votes was 5. Maybe for ( k=2 ), it's 5 again? Or maybe it's higher.Alternatively, think about the problem in terms of graph theory. Each vote is a directed edge from one citizen to another, specifically to their ( k )-th closest neighbor. So, we're looking for the maximum in-degree in this graph.In such a graph, each node has out-degree 1, so the total number of edges is equal to the number of citizens. Therefore, the maximum in-degree can't exceed the total number of citizens minus 1. But we need a more precise bound.Wait, but the citizens are arranged on a plane with all pairwise distances distinct. So, maybe there's a geometric constraint on how many times a single point can be the ( k )-th closest neighbor to others.I recall something called the "k nearest neighbor" graph, where each node is connected to its ( k ) nearest neighbors. But in this case, it's the reverse: each node votes for its ( k )-th nearest neighbor, so it's like a "k-th nearest neighbor" graph.I wonder if there's a known result about the maximum in-degree in such a graph. Maybe not, but perhaps some geometric arguments can be made.Let me think about the dual problem. For a given point ( O ), how many points can have ( O ) as their ( k )-th closest neighbor.To maximize the number of votes for ( O ), we need as many points as possible where ( O ) is their ( k )-th closest neighbor.So, for each such point ( P ), ( O ) must be the ( k )-th closest to ( P ). That means, for each ( P ), there are exactly ( k-1 ) points closer to ( P ) than ( O ), and all other points are farther away.So, to arrange this, we need to place multiple points ( P_1, P_2, ldots, P_m ) such that for each ( P_i ), there are ( k-1 ) points closer to ( P_i ) than ( O ), and all other points are farther.How can we arrange these ( P_i )s?One idea is to place ( O ) at the center and arrange the ( P_i )s around ( O ) in such a way that each ( P_i ) has ( k-1 ) points closer to it than ( O ).Wait, but if all ( P_i )s are around ( O ), then their distances to ( O ) can be controlled. Maybe arrange them in concentric circles or something.Alternatively, think about placing ( O ) and then arranging clusters of points around ( O ), each cluster containing ( k ) points. Each cluster is arranged such that within the cluster, the points are close to each other, but the entire cluster is far enough from ( O ) that ( O ) is the ( k )-th closest neighbor to each point in the cluster.Wait, let me elaborate. Suppose we have multiple clusters, each containing ( k ) points. Each cluster is placed at a certain distance from ( O ). Within each cluster, the points are arranged such that each point in the cluster has the other ( k-1 ) points in the cluster as their closest neighbors, and ( O ) is their ( k )-th closest neighbor.To achieve this, the distance between points within a cluster must be less than the distance from the cluster to ( O ). Also, the distance between different clusters must be greater than the distance from a cluster to ( O ), to ensure that points in different clusters don't interfere with each other's ( k )-th closest neighbor.But how many such clusters can we have? If we have too many clusters, the distance between clusters might become smaller than the distance from a cluster to ( O ), which would mess up the ordering.Alternatively, arrange the clusters in such a way that the distance between any two clusters is greater than the distance from any cluster to ( O ). This way, each point in a cluster only interacts with its own cluster and ( O ).But how many clusters can we fit around ( O ) without their mutual distances becoming too small?This seems similar to sphere packing in 2D, where we want to place as many clusters (which are like points) around ( O ) such that the distance between any two clusters is greater than some minimum distance.But I'm not sure about the exact number.Wait, maybe another approach. Let's consider that for each cluster, we need to place ( k ) points such that each point's ( k )-th closest neighbor is ( O ). So, each cluster contributes ( k ) votes to ( O ).If we can arrange ( m ) such clusters, then ( O ) would receive ( m times k ) votes.But how large can ( m ) be?In the plane, how many non-overlapping clusters can we arrange around ( O ) such that the distance between any two clusters is greater than the distance from a cluster to ( O ).This is similar to placing points on a circle around ( O ), each at a distance ( d ), and ensuring that the angular separation between any two clusters is large enough so that the distance between clusters is greater than ( d ).The distance between two clusters separated by an angle ( theta ) is ( 2d sin(theta/2) ). To have this distance greater than ( d ), we need ( 2 sin(theta/2) > 1 ), which implies ( sin(theta/2) > 1/2 ), so ( theta/2 > 30^circ ), hence ( theta > 60^circ ).Therefore, the angle between any two clusters must be greater than ( 60^circ ). Since a full circle is ( 360^circ ), the maximum number of clusters we can place is ( lfloor 360^circ / 60^circ rfloor = 6 ).Wait, but if each cluster takes up more than ( 60^circ ), then we can fit at most 5 clusters, because ( 5 times 60^circ = 300^circ ), leaving ( 60^circ ) unused. Or maybe 6 clusters, each separated by exactly ( 60^circ ), but then the distance between clusters would be exactly ( d ), which is equal to the distance from the cluster to ( O ). But we need the distance between clusters to be greater than ( d ), so we can't have exactly ( 60^circ ) separation.Therefore, the maximum number of clusters is 5, each separated by more than ( 60^circ ). So, with 5 clusters, each contributing ( k ) votes, ( O ) would receive ( 5k ) votes.Is this the maximum? Let me see.If we try to add a 6th cluster, the angular separation between some two clusters would have to be less than or equal to ( 60^circ ), which would make their distance less than or equal to ( d ), violating the condition that the distance between clusters must be greater than ( d ). Therefore, 5 clusters are the maximum.So, in this configuration, ( O ) receives ( 5k ) votes. Is this the maximal possible?Wait, but in the case of ( k=1 ), we saw that the maximum number of votes was 5, which fits into this formula as ( 5 times 1 = 5 ). So, this seems consistent.Therefore, generalizing, the maximal number of votes a citizen can have is ( 5k ).But let me think if there's a way to get more than ( 5k ) votes. Maybe by arranging the clusters not just around ( O ), but in some other configuration.Alternatively, maybe arranging multiple layers of clusters, each layer contributing ( 5k ) votes. But wait, if we have multiple layers, the distance from the outer layers to ( O ) would be greater, so the inner layers might interfere with the outer layers' ( k )-th closest neighbor.Hmm, this seems complicated. Maybe the initial idea of 5 clusters is indeed the maximum.Another thought: in graph theory, the maximum number of edges in a graph where each node has out-degree ( k ) is ( n times k ), but in our case, we're looking for the maximum in-degree, which could be higher, but constrained by geometry.But in our geometric arrangement, we've found a way to get ( 5k ) votes. Is there a way to get more?Wait, suppose we have more than 5 clusters, but arrange them in a way that their mutual distances are still greater than their distance to ( O ). Is that possible?If we have 6 clusters, each separated by ( 60^circ ), then the distance between adjacent clusters is ( 2d sin(30^circ) = d ). So, the distance between clusters is exactly ( d ), which is equal to their distance to ( O ). But we need the distance between clusters to be greater than ( d ), so this doesn't work.If we try to place 6 clusters with more than ( 60^circ ) separation, the total angle would exceed ( 360^circ ), which isn't possible. Therefore, 5 clusters are indeed the maximum.Thus, the maximal number of votes a citizen can have is ( 5k ).**Final Answer**The maximal number of votes a citizen can have is boxed{5k}.

💡Okay, so I need to figure out how many birds Mr. Valentino has on his farm. He has chickens, ducks, and turkeys. The problem says he has 200 chickens, twice as many ducks as chickens, and three times as many turkeys as ducks. Hmm, let's break this down step by step.First, I know he has 200 chickens. That's straightforward. Now, it says he has twice as many ducks as chickens. So, if he has 200 chickens, then the number of ducks should be 2 times 200. Let me write that down: 2 × 200 = 400 ducks. Okay, so he has 400 ducks.Next, it mentions that he has three times as many turkeys as ducks. Since he has 400 ducks, the number of turkeys should be 3 times 400. Let me calculate that: 3 × 400 = 1200 turkeys. So, he has 1200 turkeys.Now, to find the total number of birds on the farm, I need to add up the number of chickens, ducks, and turkeys. That would be 200 chickens + 400 ducks + 1200 turkeys. Let me add those together: 200 + 400 = 600, and then 600 + 1200 = 1800.Wait, does that make sense? Let me double-check my calculations to be sure I didn't make any mistakes. Starting with the chickens: 200. Ducks are twice that, so 2 × 200 = 400. Turkeys are three times the number of ducks, so 3 × 400 = 1200. Adding them up: 200 + 400 = 600, and 600 + 1200 = 1800. Yeah, that seems right.I think I've got it. Mr. Valentino has a total of 1800 birds on his farm.

💡First, recognize that a power function has the form ( f(x) = x^{alpha} ), where ( alpha ) is a constant.Given that the function passes through the point ( (9, frac{1}{3}) ), substitute these values into the function to find ( alpha ):[9^{alpha} = frac{1}{3}]Solve for ( alpha ) by expressing both sides with the same base. Since ( 9 = 3^2 ) and ( frac{1}{3} = 3^{-1} ), rewrite the equation:[(3^2)^{alpha} = 3^{-1}]Simplify the left side:[3^{2alpha} = 3^{-1}]Since the bases are the same, set the exponents equal to each other:[2alpha = -1]Solve for ( alpha ):[alpha = -frac{1}{2}]Now, the power function is:[f(x) = x^{-frac{1}{2}} = frac{1}{sqrt{x}}]To find ( f(25) ), substitute ( x = 25 ) into the function:[f(25) = frac{1}{sqrt{25}} = frac{1}{5}]

💡Okay, so I have this problem here about a parabola. The equation given is ( x^2 = 4y ). I need to find the ordinate (which is the y-coordinate) of a point A on this parabola such that the distance from A to the focus of the parabola is 5. The options are A: 2, B: 3, C: 4, D: 5. Hmm, let me think about how to approach this.First, I remember that for a parabola in the form ( x^2 = 4py ), the focus is at (0, p). So in this case, comparing ( x^2 = 4y ) to ( x^2 = 4py ), I can see that ( 4p = 4 ), which means ( p = 1 ). So the focus of this parabola is at (0, 1). Got that.Now, the problem says that the distance from point A on the parabola to the focus is 5. Let me denote point A as (x, y). Since it's on the parabola ( x^2 = 4y ), I can express y in terms of x: ( y = frac{x^2}{4} ). So, point A is (x, ( frac{x^2}{4} )).The distance between point A and the focus (0, 1) is given by the distance formula: ( sqrt{(x - 0)^2 + left( frac{x^2}{4} - 1 right)^2} = 5 ). So, I can set up the equation:( sqrt{x^2 + left( frac{x^2}{4} - 1 right)^2} = 5 )To make it easier, I'll square both sides to eliminate the square root:( x^2 + left( frac{x^2}{4} - 1 right)^2 = 25 )Let me simplify the term ( left( frac{x^2}{4} - 1 right)^2 ). Expanding that:( left( frac{x^2}{4} - 1 right)^2 = left( frac{x^2}{4} right)^2 - 2 cdot frac{x^2}{4} cdot 1 + 1^2 = frac{x^4}{16} - frac{x^2}{2} + 1 )So, substituting back into the equation:( x^2 + frac{x^4}{16} - frac{x^2}{2} + 1 = 25 )Combine like terms:First, ( x^2 - frac{x^2}{2} = frac{x^2}{2} )So now the equation becomes:( frac{x^4}{16} + frac{x^2}{2} + 1 = 25 )Subtract 25 from both sides:( frac{x^4}{16} + frac{x^2}{2} + 1 - 25 = 0 )Simplify:( frac{x^4}{16} + frac{x^2}{2} - 24 = 0 )To make this equation easier to handle, let me multiply every term by 16 to eliminate the denominator:( x^4 + 8x^2 - 384 = 0 )Hmm, this is a quartic equation, but it's quadratic in terms of ( x^2 ). Let me set ( z = x^2 ), so the equation becomes:( z^2 + 8z - 384 = 0 )Now, I can solve this quadratic equation for z using the quadratic formula:( z = frac{-8 pm sqrt{8^2 - 4 cdot 1 cdot (-384)}}{2 cdot 1} )Calculate the discriminant:( 64 + 1536 = 1600 )So,( z = frac{-8 pm sqrt{1600}}{2} = frac{-8 pm 40}{2} )This gives two solutions:1. ( z = frac{-8 + 40}{2} = frac{32}{2} = 16 )2. ( z = frac{-8 - 40}{2} = frac{-48}{2} = -24 )Since ( z = x^2 ), and ( x^2 ) cannot be negative, we discard the negative solution. So, ( z = 16 ), which means ( x^2 = 16 ). Therefore, ( x = pm4 ).Now, since point A is on the parabola ( x^2 = 4y ), substituting ( x^2 = 16 ) gives:( 16 = 4y )So, ( y = 4 ). Therefore, the ordinate (y-coordinate) of point A is 4.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the distance formula, squaring both sides, expanding, simplifying, substituting ( z = x^2 ), solving the quadratic, getting ( z = 16 ), then ( y = 4 ). It all seems consistent.Alternatively, I remember that in a parabola, the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. The directrix of the parabola ( x^2 = 4y ) is ( y = -1 ). So, the distance from point A to the directrix is ( y - (-1) = y + 1 ). Since this distance is equal to the distance to the focus, which is 5, we have:( y + 1 = 5 )So, ( y = 4 ). That's a quicker way to get the same answer without going through all the algebra. That confirms that the ordinate is indeed 4.So, the correct answer is C: 4.

💡First, I need to determine the total cost of the notebook in cents because the payment is made in nickels, and each nickel is worth 5 cents.The notebook costs 1.30, which is equivalent to 130 cents.Next, I'll divide the total cost in cents by the value of one nickel to find out how many nickels are needed.130 cents divided by 5 cents per nickel equals 26 nickels.Therefore, Mrs. Hilt used 26 nickels to buy the notebook.