The two pillars of counting are the Multiplication Principle (sequential independent tasks: multiply their counts) and the Addition Principle (mutually exclusive alternatives: add their counts). Factorials provide the machinery: n! counts all arrangements of n distinct objects in a line. Key simplification: n!/(n-r)! avoids computing full factorials — always cancel common terms. The convention 0! = 1 ensures formulas like C(n,0) = 1 and C(n,n) = 1 hold consistently. In JEE problems, always reduce expressions before computing. For instance, 15!/(13! * 2!) = $\frac{15 * 14}{2}$ = 105 is immediate. Never attempt to evaluate factorials above 12! by brute force — they exceed standard calculator limits and JEE always designs problems so cancellation is possible.