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
is a polynomial containing 8=23 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 2p 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
How to Cite this Page:
Su, Francis E., et al. "Odd Numbers in Pascal's Triangle."
Math Fun Facts.