JEE Main 2024 — Mathematics Algebra

$\begin{aligned}
& x^2(1+x)^{98}+x^3\left(1+x^{97}\right)+x^4(1+x)^{96}+\ldots \ldots \
& x^{54}(1+x)^{46}
\end{aligned}$

Coeff. of $\mathrm{x}^{70}:{ }^{98} \mathrm{C}_{68}+{ }^{97} \mathrm{C}_{67}+{ }^{96} \mathrm{C}_{66}+\ldots \ldots \ldots$x70:98C68​+97C67​+96C66​+……… $\begin{aligned}
& { }^{47} \mathrm{C}{17}+{ }^{46} \mathrm{C}{16} \
& ={ }^{46} \mathrm{C}{30}+{ }^{47} \mathrm{C}{30}+\ldots \ldots \ldots . .{ }^{98} \mathrm{C}{30} \
& =\left({ }^{46} \mathrm{C}{31}+{ }^{46} \mathrm{C}{30}\right)+{ }^{47} \mathrm{C}{30}+\ldots \ldots \ldots .{ }^{98} \mathrm{C}{30}-{ }^{46} \mathrm{C}{31} \
& ={ }^{47} \mathrm{C}{31}+{ }^{47} \mathrm{C}{30}+\ldots \ldots \ldots . .{ }^{98} \mathrm{C}{30}-{ }^{46} \mathrm{C}{31} \
& \ldots \ldots \
& ={ }^{99} \mathrm{C}{31}-{ }^{46} \mathrm{C}{31}={ }^{99} \mathrm{C}{\mathrm{p}}-{ }^{46} \mathrm{C}{\mathrm{q}}
\end{aligned}$

Possible values of $(p+q)$(p+q) are $62,83,99,46$62,83,99,46 $\Rightarrow p+q=83$⇒p+q=83