JEE Main 2024 — 9 April, Shift 1 — Question 6

$\quad \mathrm{x}^{2}(1+\mathrm{x})^{98}+\mathrm{x}^{3}\left(1+\mathrm{x}^{97}\right)+\mathrm{x}^{4}(1+\mathrm{x})^{96}+\ldots \ldots .$.$x^{54}(1+x)^{46}$

Coeff. of $\mathrm{x}^{70}:{ }^{98} \mathrm{C}_{68}+{ }^{97} \mathrm{C}_{67}+{ }^{96} \mathrm{C}_{66}+$ $\qquad$ $+{ }^{47} \mathrm{C}_{17}+{ }^{46} \mathrm{C}_{16}$ $={ }^{46} \mathrm{C}_{30}+{ }^{47} \mathrm{C}_{30}+\ldots \ldots \ldots . .{ }^{98} \mathrm{C}_{30}$

$=\left({ }^{46} \mathrm{C}_{31}+{ }^{46} \mathrm{C}_{30}\right)+{ }^{47} \mathrm{C}_{30}+\ldots \ldots \ldots . .{ }^{98} \mathrm{C}_{30}-{ }^{46} \mathrm{C}_{31}$

$={ }^{47} \mathrm{C}_{31}+{ }^{47} \mathrm{C}_{30}+\ldots \ldots \ldots . .{ }^{98} \mathrm{C}_{30}-{ }^{46} \mathrm{C}_{31}$

$={ }^{99} \mathrm{C}_{31}-{ }^{46} \mathrm{C}_{31}={ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}$

Possible values of $(\mathrm{p}+\mathrm{q})$ are $62,83,99,46$

$\Rightarrow \mathrm{p}+\mathrm{q}=83$