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In )8f) 677 Je &xad value %te Eenu @) 22 v01, 4 6 @veny _ 4 f0" T> 15 120 31 5li5) 'M exta 4C 74+2 n?-n n=%...

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In )8f) 677 Je &xad value %te Eenu @) 22 v01, 4 6 @veny _ 4 f0" T> 15 120 31 5li5) 'M exta 4C 74+2 n?-n n=%

In )8f) 677 Je &xad value %te Eenu @) 22 v01, 4 6 @veny _ 4 f0" T> 15 120 31 5li5) 'M exta 4C 74+2 n?-n n=%



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If $^{2 n} C_{4}:{ }^{n} C_{3}=21: 1$, then find the value of $n$. (1) 4 (2) 5 (3) 6 (4) 7

Okay, so this question we just want to find the oh image off given sense. First sets minus two minus 10 12 and three. And we have Fo X is equal to floor excluding my life. Okay, so we just have to perform this computation about everything. So we have ever minus two. He's gonna be floor maestro sweat. That's four divided by three, which is 1.3 repeating. This is just one if a modest one is going to be one squints. Thus when they won by three, which is just 0.3 repeating that's just zero fo zero is zero squared zero divided by three. So that's a serious side. This just for zero. This is zero. If all one is for one square to that, just one divided by three. That's his full. Is there a 0.3. Repeating was just zero f old chu for two spaces. That's 41 by three, which is one for three paintings. Us one off three is in full off three squid. That gives you nine. What? By three. That's a 43 So then our limit our image all the say s Is this going 013 to be careful, you don't put twice. Um, you don't put the same number twice more than twice into the set. Because, strictly speaking, sanitation needs to be ridden as unique numbers. So even though the 7001 is equivalent to 011 you don't write like this. You're right. In terms of organ, you numbers. So like this. No, uh, B s is equal to 012345 Wherefore s where x is well exclaimed, divide everything. Okay, so F or zero is just going to be zero squared, divided by three semesters. Zero because zero, if one is you 21 switch, that's one divided by three. This is your f off to visit would choose where that's four divided by three. This is one. If both three is then three sweat, which is nine. Divide by three. So that's his three. Divided by three. So that's also 95 by 3 53 four gives year floor 16. So for about four square is 16 divided by three. So that's gonna be, uh, 15 like that. We 5.3 repeating. So this is five. And finally fo five Easy to 25 divided by three Which should give you that. So then our sets is in 013 live in eights. Okay. See S is equal to 15 7 11 So f or F or X is easier for exploded. Three. It's one is just gonna be one squared divided by three So that's just zero f 055 squared gives 25 divided by three So that's still don't eight f 07 So seven squared gives you 49 divided by three. To that excuses, 16 f o 11 gives you 4 11 square So that's 121. What about three? So that gives you four. So then your image is then 08 16 and 40 And finally we have the s is equal to 26 10 and 14. 14 f f o x is equal to well, exclaimed the wildlife Now F 02 is in full. Also to squeak is or divided by three. So that's just one F off six is It was a 6 20 to 36. You want about three? That's just 12. Every 10 tense. What gives you 100 divided by three seconds. You 33 ever 14 is 14 square, which I am not very sure. Also start putting spread you all right. Three, which is just 65. So then you set your image effort s busy one, 12 33 and cc. Also one notes that I want to say as well is that you don't always have to write This sets in strictly ascending order or in some sort of order. You can just very easily right this whole thing as 1 33 12 65 It doesn't really matter or not so much. What matters more is a uniqueness off, but that's just a quick next.

We know that if the have excess works, my street in the inverse explosive be over for the inverse of after of 10 would be 37 the Evers of 37 would be 10 is the correct answer is 10.

Phone this problem. We're gonna be working with factorial ALS. What is a factorial? Whenever you see the exclamation mark here, it's going to mean you multiply the number by every number smaller than it until you get toe one. So four factorial is four times three times two times wine, which is 24 seven factorial, same feign. It's gonna be seven times six times, five times four times free times two times wine, which is equal to 5000 and 40. So pictorials are simple. It's just a lot of work. Zero factorial. So you might think that this is going to be zero. However, zero factorial is defined as wine. You're always going to have zero factorial be equal toe wine. The reason being that we use factorial Zina probability contacts, which is why you're doing this right now. And if I tell you there's no 10 items in a bag, pick zero of them. How many outcomes are you gonna have? Well, that's one outcome. The outcome where you don't do anything that it might feel like they're it's no outcome, but it's really one out comma. So that's why zero factorial is equal toe wine. now dividing factorial. Those this might seem like it's more work, but it's actually a little bit less work. So let's write this one out so six times, five times, four times, three times two times wine over two times wine so we can cancel out of two and a wine, and we're just left with the 654 and free, which is equal to free 60. So if bigger numbers, this is going to save a lot of time, especially if you don't have a calculator. So in this wine we have five times four times free times two times wine over two times wine times free times, two times wine. So cancel out the free to one. Now we just have five times four divided by two, which is 10 now for F, So let's start writing it out six times. Five times four times, three times two times wine over four times, three times two times wine. So six minus four factorial. That's two factorial, sometimes two times wine it and let's start cancelling stuff out for free to in one Go away and we're just left of six times five, divided by two, which is 15. Now for Geeta, this is just ah, algebra. One question that were being given here. This is getting you exposed to the notation for the binomial fear. Um, but you are just going to multiply point free times itself or times, which is 0.9 times 0.9, which is 0.8 wine. All right, for H I and J. This is something new here. We noticed there's no fraction line because these air not fractions. This is seven Jews free. We have seven objects and we're gonna pick free of them. How many different combinations can we have? There is a formula for this, which I put over here that you should have reading down any sort of n choose R sequel to end factorial over and minus R factorial times are factorial. This is why a problem f looks like the way it does case you didn't notice that. So Alexe Claude in seven and free and see what we get. So seven factorial over seven minus free factorial times free factorial. And this is equal to seven times six times five times four times three times two times 1/7, minus free 654 factorial four times free times two times wine times free times two times wine to get ready, the four free to one for free to one. And now we're now we just do whatever is left over and we get 35. So if I were going to do the same thing five factorial over five minus two factorial two factorial That is equal to five times. Four times, three times two times wine over five minus two is free. So three times two times wine times two times wine. Cancel out whatever we can and we are left with Tana. So free to zero. This is what I was talking about earlier. So free factorial over free minus zero factorial zero factorial again. This is why we need zero factorial to be equal toe one So free times two times wine over free times two times wine times wine and this gives us wine. Uh oh. This is another example of why zero factorial needs to be equal toe wine. We have three objects and we're picking zero of them. So there is one out comma because we didn't do anythings. That's the out comma. Now these last two. This these air binomial fear, um, looking equations. You're gonna be seeing a lot more stuff like this. However, once you get to this step, it's just doing algebra. It's setting up these problems. That is the hard part. So what's for choose wine? You could use the binomial fear of for this one. The end shoes. Our equation, however, think about this. You have four objects and you're picking wine. How many outcomes are there? There's four. Either pick object 123 or four. So, logically you can conclude that or choose one is equal to four, sometimes 0.2 times 0.8 cube. That doesn't make sense. You could just use the equation, though completely fine. But any sort of n choose wine is always gonna be equal to that end. And this is equal 2.4096 This last one. So again, what's 5 to 0? One What's point free? Raised to zero of power Also wine. So this is really just 0.7 race to the fifth power and that is equal 2.16807 and we are done

So here we have a table you put output. Thanks. Zero 74 1 28 21 3 53 4 56 53 6 36 7 45 8 14, 9 47. So we're asked to find F. Of three. So when X is three, what is why Axis three, Why is 53? And then we're asked to find F of X. What is our X value when Y equals one? So we find our one And our x value is too, so FA two equals one.


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