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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.
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