(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.
+26388 (a) Compute the sums of the squares of Rows 1-4 of Pascal's Triangle. That is, compute:
\(\begin{array}{|lcll|} \hline \binom10^2 + \binom11^2 &=& \binom21 =2 \\ \binom20^2 + \binom21^2 + \binom22^2 &=& \binom42 = 6 \\ \binom30^2 + \binom31^2 + \binom32^2 + \binom33^2 &=& \binom63 = 20 \\ \binom40^2 + \binom41^2 + \binom42^2 + \binom43^2 + \binom44^2 &=& \binom84 = 70\\ \hline \end{array}\)
Do these sums appear anywhere else in Pascal's Triangle?
Yes: as Central binomial coefficients
(b) Guess at an identity based on your observations from part (a). Your identity should be of the form
\(\begin{array}{|rcll|} \hline \binom{n}{0}^2 + \binom{n}{1}^2 + \cdots + \binom{n}{n}^2 &=& \binom{2n}{n} \\ \hline \end{array}\)
Yes thank you heureka, but i already got those parts i just dont know how to do part c and part d, i only know what they are i dont know how to apply them.
comitee-forming: prove this using a commitee of n people and you want to choose k from that n.
block-walking: ho many ways can you get to a certian number on pascals triangle starting from the top and going to dot by dot, each dot representing a number.
