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Combinations and Permutations
by Brian J. Armitage, Ph.D.
Ohio Biological Survey
Vol 6, No. 4, November 1998
It's confession time. I like math and statistics. There, I
said it. Now I feel much better. But, some of you are
already on guard, if not repulsed. Fear not, the load in this
article will be light. Math and statistics are merely the
tools to make a point.
I have become a student of how people confront the
world around us and how they solve problems. There are
many who toil long trying to find the one elegant solution
to any given problem. Others become very facultative in
searching out the most satisfying solution, given the
circumstances. My overall thesis is that you can predict,
but not say absolutely, how any one person will confront
a problem, how long they will take, and what ways they
will try.
This notion was derived from some training I had as a
supervisor with the Tennessee Valley Authority (TVA).
We were taught about one set, of an infinite number of
sets, of personality types and were instructed in how to
interact with each type. The four major categories for our
training session were Drivers, Analyticals, Expressives,
and Amiables. I automatically thought I would be
classified as an Analytical. However, as some of you are
well aware, I fit the Driver mode very nicely. Working
in TVA, I was also subjected to interactions with a large
number of engineers. I was impressed at their acumen
with mathematics, physical forces, and mechanical
devices. I became convinced, and still am, that some
aspects of our world can only be approached in this way.
Sending a man to the moon, building a bridge, and
calculating the size of an oil deposit are definitely areas
within the domain of an engineer or physical scientist. I
often refer to the type of thinking or approach necessary
to operate within this domain as isblack and white.ls It's
right or it's wrong with only miniscule areas of gray.
However, there are a whole array of problems and areas
of inquiry for which the black and white approach is
totally wrong. Biology and the natural world are good
examples, but by no means the only examples. This way
of thinking was first brought home to me while taking a
week-long course in experimental design during my
employment with TVA. The course instructor was Dr.
Alfred Smith (a pseudonym), the then President of the
American Statistical Union (a very heavy hitter to say the
least). About half way through the course, I tired of
hearing him give example after example about industrial
production of widgets and the like, where the amount of
variation that the data exhibited around the mean, or
average, value was very, very small. Great for widgets I declared, but what about bivalves? He stared at me
uncomprehendingly. So I proceeded to teach him some
simple facts about biological statistics, where variation of
data around the mean is often very great (often ten times
the size of the mean, or more). He was aghast! "How
could this be?" he demanded.
So we began a discussion of combinatorics. Stop! Stay
there! This is easy. Combinatorics is a time-honored
branch of mathematics concerned with counting,
arranging, and ordering. We find vestiges of it all the way
back to the Old Testament. Applications of
combinatorics are rich in both diversity and number.
Molecular biologists use it to try to determine how many
ways a gene can be positioned along a chromosome.
Psychologists use it to model the way we learn. And, the
weekend poker player .... well, you get the idea. Dr.
Smith began to feel comfortable again. I was talking his
language.
If you have two operations or units, where one consists of
two fish species and the second consists of three fish
food species, then there are 2 x 3, or 6, different ways,
of associating the fish with their food, taken one at a time.
Given the potentially large number of fish species in any
one aquatic habitat, and the even greater number of fish
food species, the possible number of interactions
between predator and prey also becomes very large.
Dr. Smith said, "So, what's the problem?" Well, I told
him that unfortunately, not all of the fish preferred all of
the fish food species equally, nor did they prefer all of the
fish food aquatic habitats or microhabitats equally, and
further, the fish often competed with each other for space
and food with varying degrees of exclusion occurring.
Then I discoursed briefly on seasonal differences,
migrations, behavioral proclivities, efficiencies of attack,
food quality, pollution, and gradients of environmental
parameters, singly and in combination. I told him that
biologists think in terms of continua and gradients,
shades of gray not black and white. His eyes began to glaze.
Biological complexity was beginning to overwhelm him.
However, I reasoned with him, some of the various
biological and environmental parameters actually
simplified things. Competition and species preferences
reduced the number of combinations possible. For
example, of a hypothetical diversity of 50 fish species in
a watershed and an average 12 species caught at any one
site within the watershed, there are over 121 billion
combinations of fish species in a given catch.
Fortunately, these large numbers of combinations never
manifest themselves. Even so, the multiplication of
probable combinations of fish by the combinations of
possible fish food organisms, could serve as an
alternative pathway to chaos. In a minor way, you can
see this every day at pizza lunch buffets. Only human
rules of semi-acceptable behavior and stomach size limits
save the day.
Then we considered another variation of counting
sequences, permutations. Here order is important. I
gave nucleotide order on a strand of DNA or RNA as an
example. I explained that order was paramount for the
three contiguous nucleotides that code for an amino
acid-specific transfer RNA. The hypothetical triplets of
AAG, AGA, and GAA (A=adenosine and G = guanine)
could cause three different amino acids to be inserted
into a chain comprising a structural protein or enzyme.
Given the number of nucleotides on a chromosome, the
number of chromosomes in a nucleus, and the frequency
and randomness of mutations and translocations, his eyes
once more glazed over. Don't worry, I said, much of the
DNA is not used and there are only 20-odd amino acids.
The rest of the week was sort of anticlimactic. Gone was
the assuredness of an industrial statistician (fortunately
this was before the full flush of Statistical Process
Control in manufacturing plants; he would have been
unapproachable then). The experimental design
examples proved very useful and we were able to see
areas of conformity in our approaches to problems.
Engineers, astronomers, physicists, mathematicians,
chemists and others live and think differently than
biologists. They often live in a black and white world.
They seek near-absolute answers verifiable to the
sixteenth decimal place. They rarely get off the "yellow
brick road" I discussed in the August newsletter, but
when they do a Nobel Prize sometimes results. They are
extremely valuable human resources for our modern
world. We couldn't easily exist without them. However,
when they ignore randomness and chance, when they
simplify to solve a problem, and when they forget about
the complexity of the natural world, they make mistakes,
and some of them are catastrophic. And, when they
apply black and white thinking to human interactions and
experiences, frustration and disappointment abound. I
love mathematics and statistics, but I'm glad my
perspectives are biological. Thinking in gradients of gray
is definitely less stressful than thinking in black and white.
I gained a lot from my class and conversations with Dr.
Smith, but there is yet much to learn. I hope our
interaction got him thinking about the magnitudes of
difference between industrial statistics and what the
natural world serves us daily.
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