The Logician and the Engineer by Nahin Paul J

Author:Nahin, Paul J.
Language: eng
Format: epub, pdf
ISBN: 9781400844654
Publisher: Princeton University Press
Published: 2013-12-31T05:00:00+00:00

The interpretation of as the number of different ways to select k correct solutions from n solutions is easy to establish. For the first correct solution we have n possibilities, for the second we have n − 1 possibilities, for the third we have n − 2 possibilities, …, for the kth correct solution we have n − (k − 1) = n − k + 1 possibilities. So, the total number of possibilities where the order of selection matters is

or, using factorial notation, . But we don’t care, in our discussion in the text, about the order of the selection for the k correct solutions, only that there are k correct solutions. Since there are (k)(k − 1) … (2)(1) = k! ways to order the k selections, then the total number of ways for selecting the k correct solutions from n solutions where order of selection is irrelevant is . Binomial coefficients occur all through mathematics, physics, and engineering (for some specific examples in physics, see my Mrs. Perkins’s Electric Quilt, Princeton University Press, 2009, pp. 261–298), and whole books have been written on them. One such book, which has been in my personal library for decades, was written by one of Shannon’s colleagues and collaborators at Bell Labs, the mathematician John Riordan (1902–1988); Combinatorial Identities, John Wiley 1968. One particularly beautiful identity comes immediately from setting x = y = 1 in the binomial theorem, giving

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