Need some help proving Pascal's Triangle properties

Question

I need to prove that the terms in each row of Pascal's Triangle increase to the middle of the row and then decrease. I'm thinking that I should compare $\displaystyle \binom{n}{k}$ and $\displaystyle \binom{n}{k+1}$ and find the condition on $k$. is this the right approach?

Answer 1

Yes it is.

The ration of both quantities is $$ \frac {n!}{k!(n-k)!}\frac {(k+1)!(n-k-1)!}{n!} = \frac {k+1}{n-k} $$ and it is $>1$ as soon as $$ {k+1}>{n-k} ;\\ k>\frac{n-1}2 $$

Answer 2

http://www.google.com/imgres?sa=X&rlz=1C1GGGE_enUS402US402&espv=210&es_sm=93&biw=784&bih=871&tbm=isch&tbnid=l6evOrv_0ghQaM%3A&imgrefurl=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPascal%27s_triangle&docid=452P88j2LBqJgM&imgurl=http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fen%2Fthumb%2F3%2F34%2FPascalsTriangleCoefficient.jpg%2F220px-PascalsTriangleCoefficient.jpg&w=220&h=220&ei=TLQfU5T8EJTjoAT4rIKABw&zoom=1&ved=0CGYQhBwwAg&iact=rc&page=1&start=0&ndsp=10&dur=0

Hint hint :). Try and use the definition of a combination to prove that n choose n/2 has the greatest number of possibilities.