Q.Find the coefficient of $x^4$ in the expansion of $(x+2)^9$.

In the expansion of $(a+b)^n$, the individual terms are labelled $t_1,t_2,\ldots,t_{n+1}$ in order, and the single formula that produces any one of them directly is the general term $t_{r+1} = {}^nC_r\,a^{n-r}\,b^r,\qquad 0\le r\le n.$ The subscript convention takes a moment to internalise: the term numbered $r+1$ (not $r$) corresponds to choosing $r$ copies of $b$, so to find, say, the 5th term one substitutes $r=4$, not $r=5$. This one formula answers three different kinds of exam question without ever writing out the full expansion. First, 'find the $k$th term' — substitute $r=k-1$ directly. Second, 'find the coefficient of $x^p$' (or of a specific power in a mixed expression) — write the general term's power of the variable as an expression in $r$, set it equal to $p$, solve for $r$, and substitute back to get the coefficient. Third, 'find the term independent of $x$' (the constant term) — the same idea, but the target power is $0$. All three question types appear together in this chapter's exercises, applied to binomials with both terms carrying powers of $x$ in the numerator and denominator (such as $\left(x^2-\dfrac1x\right)^9$ or $\left(2x^2-\dfrac{5}{x}\right)^9$), and the same formula also underlies 'reverse' problems where a stated relationship between two coefficients (e.g. one coefficient is 10 times another) is used to solve for an unknown like $k$ or $m$ inside the binomial itself.