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What is the coefficient of zn in the power series form of V1 23?...

Question

What is the coefficient of zn in the power series form of V1 23?

What is the coefficient of zn in the power series form of V1 23?



Answers

What is the coefficient of $x^{9}$ in $(2-x)^{19} ?$

He is clear, so enumerated here. So let's go over the multi no meal here. This is sea of then and sub one all the way till end Sub n is equal to in factorial over sub one factorial And so to factorial All the way till and subtheme factorial So we're interested in the coefficient for X cube. Why square See to the fifth for X plus y plus C to the 10th power. This gives us an N value of 10 in sub one value of three and stuff to value of two and then step three value of fine. So when we plug it into the equation above you get sea of 10 3 to 5 knows is equal to 2520.

Mhm now nets off exercise to 0.45. So for this exercise, we can just use a formula here. And then we got is a number which is 408,408. But here I want to mention a little more about this formula. So the term and choose and then and do to NK I often called multi nominal coefficients because they occur in the expansion of the multi nominal terms. Why one plus why do plus dots plus YK raised to the power where this sum is taken overall and one from 01 to end such that the sum of one trillion K equals. So why is this formula looks like this? Maybe you can just multiple all these terms out. And if you want to know the coefficient of this term, which means you should choose And while from these in terms and and also the coefficient, uh is it is the coefficient of y one and and two, why? To choose from these in terms and etcetera into NK uh, times y k in these terms. And this is just, uh, very similar to the partition, which means for a different object. You need to partition this object into K different groups and for each group there in one and two and do n k uh, different objects. And hence you have this formula. This is why this formula looks like this.

We know that the coefficient of this factor is just and choose from and one and two etcetera. And um Kay so in this exercise we want to compute the coefficient of X squared why to affect the answer to a tense? So it is just 17, 225 and 10. Politicals 17 factorial over two Factorial times five Factorial times 10 factorial, which he calls 408,408.

Okay, that them x to the A and why, to the nine corresponds to the term 17 8 taken from 17 three eggs to the eight and two white to the night. That's the only way that we can obtain X to the A. And why do the nine in the binomial expression? So these combinatorial number corresponds to 17 factorial over a factorial nine factorial. Now we have three to d a. That's 6561 two to the nine That's 5 12 and then we have X to the A. Why to deny. So the coefficient that we are asked to find correspond to the multiplication of thes combinatorial number, these number and this one and that is thes really, really be number.


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