(a) Compute the sums of the squares of Rows 1-4 of Pascal's Triangle. That is, compute:
(10)2+(11)2
(20)2+(21)2+(22)2
(30)2+(31)2+(32)2+(33)2
(40)2+(41)2+(42)2+(43)2+(44)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
(n0)2+(n1)2+⋯+(nn)2= 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.
+26397 (a) Compute the sums of the squares of Rows 1-4 of Pascal's Triangle. That is, compute:
(10)2+(11)2=(21)=2(20)2+(21)2+(22)2=(42)=6(30)2+(31)2+(32)2+(33)2=(63)=20(40)2+(41)2+(42)2+(43)2+(44)2=(84)=70
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
(n0)2+(n1)2+⋯+(nn)2=(2nn)
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.
