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Using Exercises 13-16, predict the largest binomial coefficient in the expansion of $(x+y)^{n}.$...

Question

Using Exercises 13-16, predict the largest binomial coefficient in the expansion of $(x+y)^{n}.$

Using Exercises 13-16, predict the largest binomial coefficient in the expansion of $(x+y)^{n}.$



Answers

In Exercises 1–4, expand the binomial using a calculator to find the binomial coefficients.
$$(a+b)^{6}$$

Okay, so we want to use binomial coefficients to expand this binomial expression on. The way we're gonna do that is we're gonna use our calculators and we're going Teoh, start at zero. And because our Experiment seven, we're gonna start at zero and go all the way up to seven. Testing each combination with seven. So we're gonna test 012 three always. Okay, so let's start with the 1st 1 When you type in zero combination seven into your calculator, you should get one. And when you do it with one, you should get seven and two combinations. Seven should give you 21 three Combination seven should give you 35 four combination seven should give you 35 five. She gave me 21 with six. That should give you seven. And with seven, it should give you one. Okay, so this bottom er right here is our aren't They are our binomial coefficients that will be using. Okay, now we want to go ahead and start riding out our polynomial. So the first thing we're gonna do is we'll do the exes first. We'll start with X two with seven, and each term will take the exponents down by one until we get to zero. So starting next to the seventh X to the sixth, next to the fifth. And so one square well s X to the one is just X waas and X to the zero is one. But we know that there's going to be something else that goes here. Um, so I'm not gonna write anything. Well, I'll just write a one so that we know that there is a one in that room just in case. Okay, Now let's do our why terms at the beginning, we're going to start with why to the zero, which is just one. And since, for multiplying by X the seventh, we don't have to ride anything. We're going to go up in the exponents until we reach wide of the Senate. So here's why does zero. It's just one. Here is why to the first and why did the first is just why why squared? Why cubed? Okay, and then we'll just keep going up, Okay. And here's where I said where we really didn't need the one because we're gonna have a why there because one times why to the seventh is just white and seven. Okay, now we can go ahead and put in our coefficients, so we'll start with the 1st 1 One goes here one times x 07 It's just extra seven, so we don't even have to write it down. Okay, seven goes next. Um, it goes on the second term. 21 goes on the next term, 35 on the next term, 35 on the next turn 21 on the next term, seven on the next term and one on the last term. But, well, we won't write anything because one times White of the seventh is again just widened. Set. Okay, so this pulling on me right here is are fully expanded version of this, Diego.

Okay, So to expand X plus wide of the 10th we're going to use the binomial theorem, so we're gonna have 10 to 0 extent. The 10 plus 10 choose one x to the nine times why plus 10. Choose to exit eight by squared plus 10. Choose three x to the seventh. Why cubed plus 10. Choose four x to the sixth. Why did the fourth plus 10 cheese five x to the fifth? Why did you think plus 10 to 6 x to the fourth white six plus 10 to 7 x to the third part of the seven, plus a 10 shoes eight X squared? Why did the eighth plus 10 choose nine x times wider than nine plus 10? Choose 10. Why to the temp. And I'm used my calculator to figure out all of these coefficients, so we'll get X to the 10 plus 10 x 10 tonight. Why plus 45 x to the eighth I squared plus 1 20 ext seven Were cute plus 2 10 x +26 Why did the fourth puts too? To stuff it? Why did they fit? Plus 2 10 extra? The fourth white of the six was 1 20 x cubed wireless of UM plus 45 x squared wide of the eight was 10 x wide of the nine plus. Why? Why to the 10.

So we're gonna find the coefficient of X to the fifth wide of the eight of this binomial expose. Why? To the 13. So we know that our truce function out in front of this term will have 13 at the top. And then our bottom number will be eight. Since the bottom number always matches the exponents of the second the second variable here. So we'll have 15 choose eight and then we can just solve that. Their country's eight is B 12. Look, 12 87. So this is our term, and this is our coefficient.

So we want to find the X of the seventh term of X minus three to the 11. So we know that we'll have X to the seventh in there, and we'll have. It'll be months wide by negative three, since that's our second term in our, uh, binomial. And the, uh, the exponents for negative three has to be four, since the exponents always have to add up to 11. So 11 minus seven is four. So we know that this has to be four and then, um, our choose function. We know it'll have 11 and it will be four since ah, it'll be 11. Choose for Since this, uh, this bottom number always matches the exponents of the second the second variable here so we can solve this. 11 truce for is 3 30 times X to the seven. Negative. Three to the fourth will be 81. So we'll have 330 times. 81 is 26,000, 700 and 30. So this is our code is our co fish in here? 26,730


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