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Unformatted text preview: 3.1. PERMUTATIONS 85 Houses 7 Cats 49 Mice 343 Wheat 2401 Hekat 16807 19607 The following interpretation has been suggested: there are seven houses, each with seven cats; each cat kills seven mice; each mouse would have eaten seven heads of wheat, each of which would have produced seven hekat measures of grain. With this interpretation, the table answers the question of how many hekat measures were saved by the cats’ actions. It is not clear why the writer of the table wanted to add the numbers together. 1 One of the earliest uses of factorials occurred in Euclid’s proof that there are infinitely many prime numbers. Euclid argued that there must be a prime number between n and n ! + 1 as follows: n ! and n ! + 1 cannot have common factors. Either n !+1 is prime or it has a proper factor. In the latter case, this factor cannot divide n ! and hence must be between n and n ! + 1. If this factor is not prime, then it has a factor that, by the same argument, must be bigger than n . In this way, we eventually reach a prime bigger than n , and this holds for all n . The “ n !” rule for the number of permutations seems to have occurred first in India. Examples have been found as early as 300 B.C. , and by the eleventh century the general formula seems to have been well known in India and then in the Arab countries....
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This note was uploaded on 09/15/2009 for the course SCF scf taught by Professor Scf during the Spring '09 term at Indian Institute Of Management, Ahmedabad.
- Spring '09