Interesting proof

[steinmasta]

I've been laboring over this problem for about three hours now, and I've resorted to asking a kind person how to solve this. For this problem, SUM=summation and n%j=binomial coefficient. So, SUM j=0->n (n%j)=(n%0+...+n%n)=2^n. We've tried figuring it out through induction and just straight algebraic manipulation but the group I've been working with hasn't a clue. We're in an honors calculus course where we're proving the fundamentals of numbers and proving the properties of fields and rings. Any insight would be great.

[Epsilon=One]

I've been laboring over this problem for about three hours now, and I've resorted to asking a kind person how to solve this. For this problem, SUM=summation and n%j=binomial coefficient. So, SUM j=0->n (n%j)=(n%0+...+n%n)=2^n. We've tried figuring it out through induction and just straight algebraic manipulation but the group I've been working with hasn't a clue.Nor do I? Apparently, I'm missing something in your notation. (Or, my ability . . .) I'm as interested in the answer to your problem as you are.

We're in an honors calculus course where we're proving the fundamentals of numbers and proving the properties of fields and rings. Any insight would be great.I'm afraid that my insight concerning your problem isn't that "great"; however, your course objective "proving the fundamentals of numbers" piqued my interest.

Does your class consider that numbers are contrivances or a construct of Nature?

What does your class consider the fundamental proof of the number One, "1"?

And, does your class consider Zero, "0," as a number or merely as a placeholder?

[HallsofIvy]

The reason they are CALLED "binomial coefficients" is that your "n%j" (more commonly represented as nCj) is the coefficient of x^jy^(n-j) in the expansion of (x+ y)^n. What happens if you set x= y= 1?