The general term T_{r+1} = C(n,r)*x^{n-r}*y^r solves three main problem types:

Type 1 -- Coefficient of x^k: In (f(x))^n, write the general term, set the power of x equal to k, solve for r, and compute the coefficient. For products like (1+x)^m*(1-x)^n, either multiply out using convolution or simplify the product first (e.g., (1-x^2)^n).

Type 2 -- Term independent of x: In (x^a + b/x^c)^n, the general term has x^{a(n-r)-cr}. Setting this to 0 gives r = an/(a+c). If r is not a non-negative integer at most n, no independent term exists.

Type 3 -- Rational terms in irrational expansion: In (p^{1/a} + q^{1/b})^n, the general term has p^{(n-r)/a}*q^{r/b}. For the term to be rational, a|(n-r) and b|r simultaneously. The number of valid r values gives the count of rational terms.

For (a-b)^n, the sign alternation (-1)^r in T_{r+1} is the most commonly forgotten factor. In (ax^p + b/x^q)^n, the constants a^{n-r}*b^r multiply C(n,r) and must not be ignored when computing the coefficient.

JEE pattern: Problems often combine two of these types, such as "find the term independent of x in (sqrt(x) + k/x^2)^n" which requires both the power equation and coefficient computation. Products of binomials like (1+x)^m*(1+1/x)^n are converted to a single expression before applying the general term.