JEE Main 2024 — 5 April, Shift 1 — Question 24

$\left(x^{1}+x^{2} \ldots .+x^{6}\right)^{4}$

$\mathrm{x}^{4} \cdot\left(\frac{1-\mathrm{x}^{6}}{1-\mathrm{x}}\right)^{4}$

$x^{4} \cdot\left(1-x^{6}\right)^{4} \cdot(1-x)^{-4}$

$\mathrm{x}^{4}\left[1-4 \mathrm{x}^{6}+6 \mathrm{x}^{12} \ldots.\right]\left[(1-\mathrm{x})^{-4}\right]$

$\left(x^{4}-4 x^{10}+6 x^{16} \ldots.\right)(1-x)^{-4}$

$\left(x^{4}-4 x^{10}+6 x^{16}\right)\left(1+{ }^{15} C_{12} x^{12}+{ }^{9} C_{6} x^{6} \ldots.\right)$ $\left({ }^{15} \mathrm{C}_{12}-4 \cdot{ }^{9} \mathrm{C}_{6}+6\right) \mathrm{x}^{16}$

$\left({ }^{15} \mathrm{C}_{3}-4 \cdot{ }^{9} \mathrm{C}_{6}+6\right)$

$=35 \times 13-6 \times 8 \times 7+6$

$=455-336+6$

$=125$