nb
I'm troubled by this.
1) You start with 'x' but provide none.
2) You have the right index for a moment, but then lose it.
3) Did you separate a and b just to confuse yourself?
4) You are asked for the coefficient. Why give the entire term?
Would you be about the business of just guessing?
The first term has C(10,0). The second C(10,1)...The sixth has C(10,5).
The first term has (ab)^10. The second (ab)^9...The sixth has (ab)^5.
It wouldn't hurt to write out the entire expansion, at least once, just so you can see it all.
I really dislike this question. I was able to have the sort of question excluded from a major contest paper.
Who is to say what the fifth term is?
$\displaystyle \sum\limits_{k = 0}^N { {{N} \choose {k}} x^k y^{N - k} } = \sum\limits_{k = 0}^N { {{N} \choose {k}} x^{N - k} y^k } $.
We might be able to ask for the middle term if N is even.