(a) Compute the sums of the squares of Rows 1-4 of Pascal's Triangle. That is, compute: 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 (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 only need help with part C!
by the way my equation is $C(n,0)^2$,$C(n,1)^2$,$C(n,2)^2$...$C(n,n)^2$=$C(2n,n)$
This might help you. Here are the first 10 rows of Pascal's Triangle and some rules about the rows of the Triangle:
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
The sum of the nth row is 2^n.
The sum of rows 0 through n is 2^(n + 1) - 1.
The generating function of the nth row is (x + 1)^n.
(a)
Compute the sums of the squares of Rows 1-4 of Pascal's Triangle.
Central binomial coefficients: \(\dbinom{2n}{n}\)
see: http://oeis.org/search?q=1%2C2%2C6%2C20%2C70&sort=&language=english&go=Suche
\(\begin{array}{|rcll|} \hline \large{\sum \limits_{k=0}^{n} \dbinom{n}{k}^2 = \dbinom{2n}{n} }\\ \hline \end{array}\)