In class we saw that the sum of the entries of row of Pascal's Triangle is . In this problem we investigate the sums of the squares of the entries of row of Pascal's Triangle.
(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 using your results from part (a).
(c) Prove your identity using a committee-forming argument.
(d) Prove your identity using a block-walking argument.
I know there is a question on this, but i need help with c and d
