How many are there? Combinatorics answers counting questions: how many -character passwords can be built with a given alphabet? In how many ways can representatives be chosen from a class of ? How many handshakes are there in a room with people? The answer requires enumerating, but without listing one by one: one uses formulae that exploit the structure of the problem.
The chapter introduces the fundamental counting principle, the factorial and the four fundamental formulae (simple arrangements and arrangements with repetition, simple combinations and combinations with repetition), the binomial coefficient with its properties and Tartaglia’s triangle, and finally Newton’s binomial theorem. The common thread is an extremely powerful mnemonic scheme, the problem of the slips: every problem is recast as drawing objects from an urn of , distinguishing whether the objects are numbered or blank and whether the same object can be drawn again or not.
Sections
- The fundamental counting principle
- Permutations, arrangements, combinations
- Binomial coefficient: properties
- Newton’s binomial theorem