+53 (a) Compute the sums of the squares of Rows 1-4 of Pascal's Triangle. That is, compute:
\(\binom10^2 + \binom11^2\)
\(\binom20^2 + \binom21^2 + \binom22^2\)
\(\binom30^2 + \binom31^2 + \binom32^2 + \binom33^2\)
\(\binom40^2 + \binom41^2 + \binom42^2 + \binom43^2 + \binom44^2\)
Do these sums appear anywhere else in Pascal's Triangle?
(b) Guess at an identity based on your observations from part (a). Your identity should be of the form
\(\binom{n}{0}^2 + \binom{n}{1}^2 + \cdots + \binom{n}{n}^2 = \text{ something}.\)
(You have to figure out what "something" is.) Test your identity for \(n=1,2,3,4\) using your results from part (a).
(c) Prove your identity using a committee-forming argument.
(d) Prove your identity using a block-walking argument.
This is a repost of http://web2.0calc.com/questions/in-class-we-saw-that-the-sum-of-the-entries-of-row-n-of-pascal-s-triangle-is-2-n-in-this-problem-we-investigate-the-sums-of-the-squares which is totally dead.
Source: AoPS
If you are going to answer please provide your reasoning for Parts (c) and (d)
+118687 Hi Itzcubez,
I have no idea what c or d even mean. Sorry. Here's the first bit.
(a) Compute the sums of the squares of Rows 1-4 of Pascal's Triangle.
\(1+1=2\\ 1+4+1=6\\ 1+9+9+1=20\\ 1+16+36+16+1=70\\ \binom{n}{0}^2 + \binom{n}{1}^2 + \cdots + \binom{n}{n}^2 = \frac{(2n)!}{(n!)^2} \)
Do these sums appear anywhere else in Pascal's Triangle?
\( \binom{n}{0}^2 + \binom{n}{1}^2 + \cdots + \binom{n}{n}^2 = \frac{(2n)!}{(n!)^2}\)
(b) Guess at an identity based on your observations from part (a). Your identity should be of the form
(You have to figure out what "something" is.) Test your identity for using your results from part (a).
You can do the testing but yea it works :)
(c) Prove your identity using a committee-forming argument.
(d) Prove your identity using a block-walking argument.
This is a repost of http://web2.0calc.com/questions/in-class-we-saw-that-the-sum-of-the-entries-of-row-n-of-pascal-s-triangle-is-2-n-in-this-problem-we-investigate-the-sums-of-the-squares which is totally dead.
Source: AoPS
If you are going to answer please provide your reasoning for Parts (c) and (d)
Oh I see this is one of Mellie's old questions - She asked some great questions :)
