The Binomial Theorem-Particle Distributions

To describe particle distributions, we will use nomenclature relating to 

Measuring particle size and growing single crystals 

This is the BINOMIAL THEOREM. By Taylor Expansion, we can find the total probability, P, for items, n, taken r at a time as  

One will immediately recognize this as a form of the BOLTZMANN equation, or the GAUSSIAN LAW. We can modify this equation and put it into a form more suitable for our use by making the following definitions. 

we define a mean (average) size of particles in the distribution as d, given as  

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