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From theFun Factfiles, here is a Fun Fact at the Advanced level:

Odd Numbers in Pascal's Triangle

Figure 1

Pascal's Triangle has many surprising patterns and
properties. For instance, we can ask:
"how many odd numbers are in row N of Pascal's Triangle?"
For rows 0, 1, ..., 20, we count:

Since the coefficients
of these polynomials are equal [mod 2], using the
binomial theorem we see that
(10 CHOOSE k) is odd for
k = 0, 2, 8, and 10; and it is even for all other k.
Similarly, the product

(1+x)^{11}
= (1+x^{8})(1+x^{2})(1+x^{1})
[mod 2]

is a polynomial containing 8=2^{3} terms, being
the product of 3 factors with 2 choices in each.

In general, if N can be expressed as the sum of
p distinct powers of 2, then (N CHOOSE k) will be odd
for 2^{p} values of k.
But p is just the number of 1's in the binary
expansion of N, and (N CHOOSE k) are the numbers in
the N-th row of Pascal's triangle. QED.

For an alternative proof that does not use the binomial theorem
or modular arithmetic, see the reference.
For a more general result, see
Lucas' Theorem.

How to Cite this Page:
Su, Francis E., et al. "Odd Numbers in Pascal's Triangle."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.