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?

Guest May 24, 2019

#1**+2 **

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

Guest May 24, 2019

#2**+2 **

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.

Melody May 24, 2019

#3**+2 **

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 a_{m}a_{m-1}....a_{0 }and the representation of k in base p is b_{m}b_{m-1}....b_{0} 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 a_{m}a_{m-1}....a_{0} 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

Guest May 25, 2019

edited by
Guest
May 25, 2019

edited by Guest May 25, 2019

edited by Guest May 25, 2019