Saturday, August 4, 2012

Exploring Exponential Expansion: Combinations!

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I am fascinated, as are most people, at rates of expansion, growth and change. Albert Einstein once said (in a Swiss accent whilst tripping over an untied shoelace) that "the most powerful force in the universe is the compounding of interest."

The idea of arithmetic, geometric, exponential and logarithmic expansion, looked at singly or comparatively, give us accessible, understandable metrics about how much things change as a result of the rate or power of growth. Everyone has an interest in these things...from the little tike saving up his meager allowance for his first WonkaBar lottery ticket purchase, to the epidemiologists (mostly the ones in the shows on late night television anyway) who must estimate the potential for a virus or other threatening bio hazard to spread, the the idiot from some fourth-world nation who sends out a chain letter to a whole group of people saying to each of them that if he or she sends the letter to an additional seven people enclosing one [insert any imaginary unit of currency], he or she will become a billionaire [i.e., that he or she will receive one billion units of the imaginary units of currency] within seven weeks - or, alternatively, that if he or she (poor recipient) fails to  do as instructed (gasp!!!), that he or she will be visited by the angel of halitosis and be breathed upon until his or her skin is covered with festering boils...or worse.

One area worth exploring for all those of you involved in business and networking, or search engine optimization and link-building, is that of combinatorial analysis. This is the process through which we answer the question "How many combinations of r objects each are possible out of a total group of n objects.

The formula used to solve this type of problem is:

where n is the number of things to choose from, and you choose r of them
(No repetition, order matters)

As a practical, yet tepid and unimaginative example, let us assume that We are invited to a business power breakfast at the local Olympia Diner. If there are ten invitees in total, and each person shakes hands once with every other person at the end of the meeting (without anyone  leaving a tip for Rayette, the waitress with a prominent mole who calls everybody "hon' "), how many handshakes will there be in all?

This question, restated without the cynical editorial commentary, is simply, "How many combinations of 2 are possible out of a group of 10, in total?"

Using the above formula, we take 10!, and divide it by (10! - 2!), simplify it to 10! divided by 8!, further simplify to 10 x 9, and we obtain an answer of 90 possible handshakes.

By the way, if you need an automatic full factorial calculator just click *HERE*. I never leave the house without mine. It beats trying to figure out 123! either in my head (which is rather full) or on the back of all of my wife's best linen table napkins with my Bic.

Here's one for You:

If I own fourteen suits, and I wear two different suits during any given week, how many different combinations of suits will I be able to wear?

Here's another for You, with a slight twist:

If I publish 50 different blogs, and I reference all of the other blogs on each one of the blogs, how many total links will I have to any one given blog on each of the others, (inclusive of that blog itself)? Hmmm? And how many links will I have which link one blog to another?

Stay tuned for the answers.

Douglas E. Castle for The Braintenance Blog

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