Derivation of some formulae in combinatrics by heuristic methods

摘要

Heuristic methods are more effective for students inlearning permutations and combinations in mathematics than passive learning such as rote memorization of formulae. Two examples, n! and (2n-1)C_n, of finding new combinatorial formulae are discussed from a pedagogical standpoint. First, the factorial of n can be expressed as ∑_(k=0)~(n-1)k·k!, which can be found by a heuristic method. This expression is comparable to representations of powers of r using geometrical series. Second, the number of possible combinations with repetition of n drawings from n elements is denoted (2n-1)C_n, which can be calculated from ∑_(k=0)~(n-1)_nC_(k+1n-1)C_k. The relation ∑_(k=0)~(n-1)_nC_(k+1n-1)C_k=(2n-1)C_n can be found by a heuristic method through a corresponding problem on mapping.