# Pascal's triangle

## Primary tabs

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

## Mathematics Subject Classification

05A10 Factorials, binomial coefficients, combinatorial functions

### proof by inspection

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.

### Re: proof by inspection

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

### Re: proof by inspection

> 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 ---

### Re: proof by inspection

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

Cam

### Re: proof by inspection

> 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.

### Re: proof by inspection

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.