The Brilliant Set - 1729 Special

$\text{Motivation}:$ As soon as I saw that we had to find the sum of products of $2$ , $3$ ..... $1728$ elements of set $B$ ,the first thing that struck me was using polynomials to solve this problem,as in a polynomial,the co-efficients are like that. $$ $\text{Method}:$ We first define a monic polynomial $p(x)$ which has roots $1,2,3,.....1727,1728$ ,hence $p(x)=(x-1)(x-2)....(x-1728)\\ =x^{1728}-x^{1727}(sum\ of\ roots)+x^{1726}(sum\ of\ product\ of\ roots\ taken\\ \ two\ at\ a\ time)-....+1728!$ ,now since we require the sum of co-efficients $-$ the first two terms we put $x=-1$ as then we will get the sum!Here,we have put $x=-1$ instead of putting $x=1$ as if we did the latter,we would not get the sum as there would be some $-$ signs in between.Now,substitute $x=-1$ to get, $p(x)=1729!$ but,as said before,we have to subtract the first two terms,then we have $Answer=1729!-(1+\dfrac{1728\times 1729}{2})$ from here we can easily find the answer.And done!