Resumen de Derivation of some formulae in combinatrics by heuristic methods

Yukio Kobayashi

• 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- 1Cn, of finding new combinatorial formulae are discussed from a pedagogical standpoint. First, the factorial of ncan be expressed as ?n- 1k= 0k· k!, which can be found by a heuristic method. This expression is comparable to representations of powers of rusing geometrical series. Second, the number of possible combinations with repetition of ndrawings from nelements is denoted 2n- 1Cn, which can be calculated from ?n- 1k= 0nCk+ 1n- 1Ck. The relation ?n- 1k= 0nCk+ 1n- 1Ck= 2n- 1Cncan be found by a heuristic method through a corresponding problem on mapping.