the coeff. of x^1274 in the expansion of (x+1)(x--2)^2(x+3)^3.......(x-50)^50 is?

x^1274 is the second biggest term.                                                                                                             Biggest term is when each braccket gives maximum x ie.x^(1+2+3...50) and (1+2+3.....50)=50*51/2=1275.     So, coff.(x^1274) will be when 49 brackets give maximum power of x and one remaining bracket gives second largest power of x. If first bracket gives second largest power, then coff. of that term will be 1C0*(1). If i-th bracket gives second largest power of x, then coff. of that term will be iC(i-1)*((-1)⁽ⁱ⁺¹⁾ * i ).                                          So, total coff. will be sum of all such cases, ie.             [50C49*(-50) + 49C48*(49) + 48C47*(-48) + …... 1C0*(1) ]          which is = 1 – 2^2 + 3^2 -4^2 …... – 50^2, which you can solve by clubbing n-th and (n+1)-th term together. Just write (n+1)^2 – n^2 = n^2 + 1 +2*n – n^2, and you can cancel all square terms.