$$
\left(\frac{x}{3}+9 y\right)^{10}
$$

Answer

Need to find the middle term in the expansion of \[\Big(\frac{x}{3} + 9y\Big )^{10} \]
Since $n=10$ is even, the middle term is $\Big(\frac{n}{2} +1\Big)$th term.
Hence, $T_6$ is the middle term and it can be computed using:
\[T_{r+1} = {n \choose r} a^{n-r}b^{r}\]
\[T_6 = {10 \choose 5} \Big(\frac{x}{3} \Big)^5 \big(9 y \big)^5\]

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Algebra

NCERT Class 11 - Math

Chapter 8

Binomial Theorem

Section 2

Binomial Theorem for Positive Integral Indices

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Video Transcript

So uh today we're trying to find out a solution for this problem here. We need to find the middle term in the expansion of this particular expression that is X by three plus nine by hold, poverty in this is 1/11 standard and CRT matt's problem. All right. So uh for this be used binomial theorem. Let's quickly see what binomial theorem is. So binomial theorem for any positive integer states that any positive integer end I mean states that for a plus B hold our whole rise to end. The expansion is in the form of a summation. In terms of uh N C R A and B. The powers of a N B. Right. So A A plus beagle parent is submission. R is equal to 0 to 10. And see our a part and my nsR plus beep our are times people are all right. So, uh what is NCR here? And crs and factorial by and minus R. Factorial into our factory. So with that out of the way, we can finally look at the expansion of our given problem given problem expression here. So we need the expansion for X by three plus nine. Y whole Parton. So, simply plugging in the corresponding numbers and terms we get it is the submission R is equal to zero to end 10. Cr into X by three whole part. 10 minutes are into nine. Weapons are So uh as you can see from the number of terms in the submission that is 0-10. We can know that the number of terms is A total of 11 terms. Hence the middle term will be the term and by two plus one. So t and by two plus one will be our middle term that we need to find for this problem. So finding the sixth term and by two plus wonders. And by two plus 1, 10, 10 by 25 plus one is six. Finding sixth turn is can be done using this Formula 80 are pressed, one is equal to and C. R. A. Bar and minus R B. Para even. We can even find that in a brute force method like you know there are numbers from 0 to 10. Return an order. So which which term will it be in terms of our it will be R is equal to five. Right, so you can either use this or that. So T six is 10 C five X by three hole par five and times 95 par five. So uh you simply expand each of these. So 10 C five becomes 10 factorial into 10 minus five factorial. Which is five factorial and times five factorial and that is multiplied by X by three whole power five. So X by three whole power five. This ex parte five by three par five and nine can be written as three square. So that uh three squared power 53 power 10. So three power 10 divided by three par five will get cancelled out in the next uh steps. So that's why we wrote it this way. So that times y power five. So a simple manipulation of this multiplication and all that you can cancel out 10 factor five factorial from you know, Numerator and denominator. And you'll be left with 10 10 times nine times eight times seven times six by five factorial. And you'll be left with the multiple um three power five from this term. And then expert find by part five. If you simply multiply all those things you will get the answer to be 61236 x power if I wipe our five, this is your final answer.

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