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MathsJEE Main 22 Jan 2026, Shift 1·Q16·MCQ·Tough

Binomial Theorem · Coefficient in sum of binomial expansions

The coefficient of x48x^{48} in (1+x)+2(1+x)2+3(1+x)3+…+100(1+x)100(1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + \ldots + 100(1 + x)^{100} is equal to:
  1. (1)100⋅100C49−100C50100\cdot{}^{100}C_{49} - {}^{100}C_{50}
  2. (2)100C50+101C49{}^{100}C_{50} + {}^{101}C_{49}
  3. (3)100⋅100C49−100C48100\cdot{}^{100}C_{49} - {}^{100}C_{48}
  4. (4)100⋅101C49−101C50100\cdot{}^{101}C_{49} - {}^{101}C_{50}
We corrected a printing error in the original paper

Source printed option (4) as “100⋅101C49−100C50100\cdot{}^{101}C_{49} - {}^{100}C_{50}” (then no option is correct); corrected to the intended “100⋅101C49−101C50100\cdot{}^{101}C_{49} - {}^{101}C_{50}” (the correct coefficient and the source solution's result; printed key (4)).

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See the full 22 Jan 2026, Shift 1 paper