Strategy 1 -- Coefficient extraction: Write the general term, set the power to the target, solve for r. For products, either convolve (multiply) the expansions or simplify the product first (e.g., (1+x)^m*(1-x)^m = (1-$x^2$)^m).

Strategy 2 -- Coefficient sums: To evaluate sums involving C(n,r), identify the generating function. If the sum has weights like r, $r^2$, $\frac{1}{r+1}$, use differentiation or integration of (1+x)^n. If it has alternating signs, substitute x = -1.

Strategy 3 -- Remainder problems: Write the base as (multiple of divisor + remainder) and expand. For repeated squaring problems (like 2^{2022} mod 7), use Fermat's little theorem to reduce the exponent first.

Strategy 4 -- Counting rational/integral terms: In ($p^{1/a}$ + $q^{1/b}$)^n, count values of r where both $\frac{n-r}{a}$ and r/b are non-negative integers.

Strategy 5 -- Hockey stick and Vandermonde: Recognize sums of binomial coefficients along rows or diagonals of Pascal's triangle. These identities often simplify multi-step summations into a single binomial coefficient.

Strategy 6 -- Factorization trick: (1+x+$x^2$)^n = $\frac{(1-x^3}{(1-x)}$)^n allows conversion to a product of two simpler series. Similarly, (1+x)^n*(1-x)^n = (1-$x^2$)^n simplifies products.

Common traps: (1) Missing (-1)^r in (a-b)^n expansions. (2) Confusing "sum of coefficients" (put x=1) with "sum of binomial coefficients" (2^n). (3) Forgetting convergence condition |x| < 1 for generalized binomial. (4) Miscounting the term number ($T_{r+1}$, not $T_r$, has coefficient C(n,r)).