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Pascal's triangle

Synonym: 
Tartaglia's triangle
Type of Math Object: 
Topic
Major Section: 
Reference

Mathematics Subject Classification

05A10 no label found

Comments

Looked at the right way, this fact that the sums of entries in
rows are powers of two becomes obvious. By definition, each
number in a row except the two 1's at the ends is gotten by adding
the the two elements immediately above it. So write the sum of
elements in a row, then replace the elements in the middle by
their expression as sums. For instance, rewrite

1 + 5 + 10 + 10 + 5 + 1

as

1 + 1 + 4 + 4 + 6 + 6 + 4 + 4 + 1 + 1 .

Note that each number appears twice in the latter sum. Hence,
the sum of numbers in a row equals twice the sum of the numbers in
the preceding row. Since the sum of the numbers in the top row
is 1 = 2^0, this means that the sums of the rows are successive
powers of two.

Isn't that just the induction argument alluded to in the entry?

Another slick proof is to do it combinatorially: The k-th entry in the n-th row is just the number of ways of choosing k out of n objects. The sum of all the entries in the row is the number of ways of choosing any subset of the n objects, of which there are 2^n.

Cam

> Isn't that just the induction argument alluded to in the entry?

Pretty much, just presented in a way which can be easily grasped
by a fifth-grader. Personally, I don't see what is so inelegant
about the induction proof --- especially as presented here, it
follows rather immediately from the construction of the triangle.
For me, it is less elegant to first prove the binomial theorem,
but this is a matter of taste so it is not like either me or Koro
are right or wrong. At any rate, nice to have three proofs ---
now, I have no doubts about this result ;)

Sure -- I certainly wasn't arguing that your proof (or the more formal induction method) was any less elegant than any other method.

Cam

> Sure -- I certainly wasn't arguing that your proof (or the
> more formal induction method) was any less elegant than any
> other method.

The comment wasn't directed at you --- rather it is a response
to what the entry says --- "This can be easily proved by
induction, but a more elegant proof goes as follows:". However,
as I said earlier, elegance is a matter of personal taste, so
you, Koro, and I are equally entitled to our opinions
as to elegance.

correction: it should have said

"you Koro, Matte, and I" since Matte is a coauthor.

In addition to being able to search fora, it would be
nice to be able to edit posts.

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