There are four even integers in the top five rows of Pascal's Triangle. How many even integers are in the top 10 rows of the triangle?
Just count them:
1 = R 0
1 1 = R 1
1 2 1 = R 2
1 3 3 1 = R 3
1 4 6 4 1 = R 4
1 5 10 10 5 1 = R 5
1 6 15 20 15 6 1 = R 6
1 7 21 35 35 21 7 1 = R 7
1 8 28 56 70 56 28 8 1 = R 8
1 9 36 84 126 126 84 36 9 1 = R 9
1 10 45 120 210 252 210 120 45 10 1 = R 10
Thanks Guest that is a good answer.
I just counted them too but I did not need to work out all the numbers.
I drew it in the shape of the normal triangle but insead of putting the actual number I just put o for odd and e for even.
If 2 odds are added the answer will be even
If 2 evens are added the answer will be even
If one even and one odd are added the answer will be odd.
n choose k is NOT divisible by some prime number p if and only if:
\(\{\frac{k}{p^i}\} \leq \{\frac{n}{p^i}\} \) ({x} is the fractional part of x) for all \(i\in\mathbb{N}\)
In other words, if the representation of n in base p is amam-1....a0 and the representation of k in base p is bmbm-1....b0 then n choose k is NOT divisible by p if and only if \(\forall 0\leq i \leq m\enspace b_i \leq a_i\)
conclusion: if the representation in base p of n is amam-1....a0 then there are exactly \(\prod_{i=0}^{m} (a_i+1)\) values of k between 0 and n such that n choose k is NOT divisible by p
EDIT: here's a useful stack exchange thread