JEE Main 2024 — 31 January, Shift 2 — Question 17

${ }^{6} \mathrm{C}_{\mathrm{m}}+2\left({ }^{6} \mathrm{C}_{\mathrm{m}+1}\right)+{ }^{6} \mathrm{C}_{\mathrm{m}+2}>{ }^{8} \mathrm{C}_{3}$

${ }^{7} \mathrm{C}_{\mathrm{m}+1}+{ }^{7} \mathrm{C}_{\mathrm{m}+2}>^{8} \mathrm{C}_{3}$

${ }^{8} \mathrm{C}_{\mathrm{m}+2}>^{8} \mathrm{C}_{3}$

$\therefore \mathrm{m}=2$

And ${ }^{n-1} P_{3}:{ }^{n} P_{4}=1: 8$

$\frac{(n-1)(n-2)(n-3)}{n(n-1)(n-2)(n-3)}=\frac{1}{8}$

$\therefore \mathrm{n}=8$

$\therefore{ }^{n} P_{m+1}+{ }^{n+1} C_{m}={ }^{8} P_{3}+{ }^{9} C_{2}$

$=8 \times 7 \times 6+\frac{9 \times 8}{2}$

$=372$