# Pascal's Triangle and Binomial Coefficients

Prove that if you move straight down in Pascal's Triangle, visiting every other row, then the numbers are increasing.

• so you want to prove $\binom{n}{k}\leq \binom{n+1}{k}$? – Jorge Fernández Hidalgo Feb 17 '17 at 0:34
• Wouldn't it be (n+2 k+1) because it's every other row – JanoyCresva Feb 17 '17 at 0:37
• I'm not sure, I'm confused how you would even derive that. – JanoyCresva Feb 17 '17 at 0:39

...prove that $\frac{n!}{2(n/2)!}$ for $2 | n$ is increasing as $n$ increases? Or down starting at arbitrary $k$ ans $n$? The latter is also fairly trivial:

$\binom{n+2}{k+1} = \binom{n+1}{k} + \binom{n+1}{k+1} = \bigg(\binom{n}{k-1} + \binom{n}{k}\bigg) + \binom{n+1}{k+1} > \binom{n}{k}$

• Is this enough to fully prove it? – JanoyCresva Feb 17 '17 at 1:00
• Yes, so long as you have proved that $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$. This of course implies, more strongly, that every element of each row of Pascal's triangle is larger than every element of the previous row, which immediately proves your statement. – dasaphro Feb 17 '17 at 4:36