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(10 Fi which third You F i8 first L runners sSconapla (40 !N are # finish are shmpetie gop 'OMJ 0 How marathon many diclerint first...

Question

(10 Fi which third You F i8 first L runners sSconapla (40 !N are # finish are shmpetie gop 'OMJ 0 How marathon many diclerint first

(10 Fi which third You F i8 first L runners sSconapla (40 !N are # finish are shmpetie gop 'OMJ 0 How marathon many diclerint first



Answers

Footrace There are 50 runners in a race. How many ways can the runners finish first, second, and third?

All right, So to begin with, we were told that there are three people that were concerned with in this race that there is a first place winner who has traveled one mile. And that's the first place. Uh, cool. Brenner finishes 10 feet ahead of the second place, and the second place at that moment is 10 feet ahead of the, uh, third clicks. I would highly recommend that you kind of work through what's going on in this problem. Before you watch this video or a different points pause this video because this one really rack him. I had Iraq, my brain, to try to figure this out, and so I haven't approach. But I I would recommend that you kind of work through it as well to try your best before you get the answers and the method that I have. So the way that I have crunched this was I said, All right, so the distance that the second things person has gone so far is 5270 feet, because one mile is 5280 and that means that the third place is 5260 the, um, time that they have both traveled up until this point, this tea is I'm gonna put t here and here. That means that the rate at which they were running and that they will continue to run is the distance divided by time. So then I have to think about the relationships that are here because our goal is for the second place person to actually get to 5280. And so if they get to 5280 then that's gonna be equal to the rate at which they run 5270 over tea multiplied by the time that they ran already, plus an additional time. And I'm gonna call that additional time Delta team because in calculus and many other context, Delta means change. So this is the change in time. So this is an expression or an equation that shows the relationship between the time that we care about and the rate at which the second place runner is running. So then what I would dio is multiply everything by t because I'll get 5280 t equals and then I'm going to distribute the 5270 since the tea is no longer there. So this goes to both terms 5270 T plus 5270 Delta T Then if you subtract, you're going to get 10 t, which actually makes sense if you if he runs for 10 more feats to get there all rights. So what we have? No, we can divide both sides By 10. It's a t will equal of 527 multiplied by Delta T. What this basically means is that the time that it took the second place runner to get to this distance is 527 times. The amount is going to take this runner to go. 10 more feet hands. So then what I'm going to Dio is right, an expression for this third place person. So the third place person is going to run a certain extra distance during this time, so they're running the distance that's equal to their rates. 5260 over G multiplied by the time that they're running. That last little drug has dealt a team, and so then I can substitute the value of tea or the expression of tea here. So what that tells us is that the change in the distance that this third place runner goes in the same time that the second place runner finishes the race is going to be equal to five thousands, 260 over 5000 270. Delta team multiplied by Don't to see so that the Delta teams will cancel and we have that the don't it d the difference in the distances from when they got to this point to when the the second place runner finishes is equal to this. So then, if you substitute or plug this into a calculator, you would get the Delta Dean equal to this amount. This approximately what? I have nine 0.98 feet. Their distances Aaron Feet. So 9.98 feet. All right. The second place runner had gone 10 feet and they had been 10 feet ahead of the third place runner. So that means that in the in the time that it took the second place runner to get that extra 10 feet, the third place runner only went nine and 98 hundreds of foot, which means they lost to hundreds of a foot. Um, in in between. Originally, I was thinking, Whoa, I mean, it's it's almost the same, but because they're running is is equivalent to their do their previous rate, the 10 feet doesn't see the same. Their rates stay the same, and so based off of that, this is the distance that the third place runner ran. And so that means that there's two extra hundreds. So that means instead of just 10 feet in between these two, there's not 10 feet and to hundreds. So the distance between the second and third place runners dot, dot dot and, the doctor thought, is when the second place runner has finished. The race would be 10.2 feet, so essentially it's the same thing as 10 feet. But it because of all the the slight differences in in speed, it is slightly different. It's less than it's significantly less than an inch, Um, so it's just it's fascinating that we have to do all this work to find 2/100 of a foot

Okay, so, in this problem, you're given the information that this race is a one mile long, which is 5280 ft, and the winners already crosses the finish line. So, the winner is already done with 5280 ft. So, but but it tells you that the winner crosses the finish line 10 ft ahead of the second place runner. So that means that the second place is still going to 10 ft off from 5280. So you're going to do 5280 Subtract 10. So, that means the second runner is at 5270 ft, and then the third place runner, uh it says the winner is 20 ft ahead of the third place runner. So that means the third place runner is 20 ft behind. So, subtract 20 and that's gonna give you 5260 ft. So, we are assuming that each runner maintains the same speed throughout the race by how many feet does the second place runner beat? The third place runner? Okay. So we can look at this and know that the second place runner is beating 10 ft because you can just subtract 5270 subtract 5260. Or you can set up the ratio, you know that um the second runner is at 5270 out of 5280 total. And that's gonna be asking us, you know, if if the third place runner is at 5260 where were the third place runner is going to be? Okay? So if you cross multiply and you solve for X. X. Is going to be approximately 5269.98 So this tells you that uh this is where the third runner is, when the second uh winner crosses the finish line. So now you know that, So now you can just go ahead and do 5280 subtract 5269.98 and the answer is still around 10 ft, so you can also set up the proportion if you want to or or you can just look at this.

Here we have eight runners and we're figuring out how many different ways there are to have a 1st 2nd and third place runner. And since order matters here, this would be a permutation situation. So we have eight p three for the eight runners total and choosing three of them. And so, according to our permutation formula, that would be a factorial divided by eight minus three factorial, which is the same as a factorial divided by five factorial. Now perhaps you're using your calculator to figure this out, but just in case you're multiplying it out my hand, you can expand a factorial and you get eight times seven times, six times five factorial and that's over five factorial, so you can cancel the five factorial tze and all you have left is eight times seven times six and that gives you 836. And the answer that goes with that is J

Since the object here, the place of the order in this triathlon cannot be repeated in order absolutely matters here when you are um identifying 1st, 2nd and 3rd place. This is a permutation problem. So using the permutation formula and identifying that we have 100 people competing and we're going to order three of them. We want to find 100 per mutate three. So it's 100 factorial over 100 minus three factorial or 100 factorial over 97 factorial. 100 factorial is 100 times 99 times 98 times 97 factorial over 97 factorial. So that we can just cancel those out and it's 100 times 99 times 98 which is equal to 970,000 200 different permutations.


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