Derivation of some formulae in combinatrics by heuristic methods

• Autores: Yukio Kobayashi
• Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 46, Nº. 3, 2015, págs. 469-476
• Idioma: inglés
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• Resumen

  • 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.