Simplify the expression

x²⁶ - x²⁵(a+b+c+...+z) + x²⁴(ab+ac+...+az+bc+...+bz+...+yz) - x²³(abc+...+abz+acd+...+acz+...+xyz) + ... + abc...z

I'm not sure, but my guess is zero.

It seems like each "part" of the expression is canceled out by the next "part":

x^26 (the first "part") is canceled out when -x^25 is multiplied by x (in the second "part").

-ax^25, -bx^25, etc (in the second "part") are canceled out when x^24 is multiplied by ax, bx, etc (in the third "part").

Get it?

So the next to last part will be x times the alphabet missing one letter (for each letter).  The 25 of those that contain an x^2 will cancel out terms from the previous part.  The only one that will remain is -abcdefghijklmnopqrstuvwxyz, which is canceled out by the final part, so the result is zero.

I know that make very little sense, but that's my guess. :)

Posted by tomarken on 2006-06-16 09:26:14