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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 already did part (a), but I still need help on parts (b), (c), and (d). Thanks for helping!

 Apr 10, 2019
 #1
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+4

Could I please have some help on this?

 

\(\begin{array}{|lrcll|} \hline n=0&\binom00^2 &=& 1 =\binom00 \\ n=1&\binom10^2 + \binom11^2 &=& 1^2+1^2 = 2 =\binom21 \\ n=2&\binom20^2 + \binom21^2 + \binom22^2 &=& 1^2+2^2+1^2 = 6 =\binom42 \\ n=3&\binom30^2 + \binom31^2 + \binom32^2 + \binom33^2 &=& 1^2+3^2+3^2+ 1^2 = 20 =\binom63 \\ n=4&\binom40^2 + \binom41^2 + \binom42^2 + \binom43^2 + \binom44^2 &=& 1^2+4^2+6^2+4^2+ 1^2 = 70 =\binom84 \\ &\ldots \\ &\dbinom{n}{0}^2 + \dbinom{n}{1}^2 + \cdots + \dbinom{n}{n}^2 &=& \dbinom{2n}{n} \\ \hline \end{array} \)

 

 

laugh

 Apr 11, 2019
 #2
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+1

thanks!

 Apr 11, 2019

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