The famous factorial again!

Multiply both sides by $10!$ .

$n = \frac{10!}{2!9!}+\frac{10!}{3!8!}+\frac{10!}{4!7!}+\frac{10!}{5!6!}= \frac{10}{2} + \frac{90}{6} + \frac{10 \cdot 9 \cdot 8}{24} + \frac{10 \cdot 9 \cdot 8 \cdot 7}{120} = 5 + 15 + 30 + 42 = 92$

multiply by 11!: $11n=\binom{11}{2}+\binom{11}{3}+\binom{11}{4}+\binom{11}{5}$ add $\binom{11}{0}+\binom{11}{1}=12$ . $11n+12=\binom{11}{0}+\binom{11}{1}+\binom{11}{2}+\binom{11}{3}+\binom{11}{4}+\binom{11}{5}$ by fundemental combo identity $11n+12=\binom{11}{11}+\binom{11}{10}+\binom{11}{9}+\binom{11}{8}+\binom{11}{7}+\binom{11}{6}$ add these two : $22n+24=\binom{11}{0}+\binom{11}{1}+\binom{11}{2}+\binom{11}{3}+\binom{11}{4}+\binom{11}{5}+\binom{11}{6}+\binom{11}{7}+\binom{11}{8}+\binom{11}{9}+\binom{11}{10}+\binom{11}{11}$ by the binomial this is just $22n+24=2^{11}=2048$ $n=\dfrac{2048-24}{22}=\boxed{92}$

Multiply by 11! So 11C2+11C3+11C4+11C5=11n i.e. 12C3+12C5=11n i.e. 12×11×10/6+12×11×10×9×8/120=11n So n=12×10/6+9×8=92