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  4. In n-1C3 + n-1C4 > nC3, then n is just greater than integer.

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Q. In $^{n-1}C_3\,+\,{}^{n-1}C_4\,>\,{}^nC_3$, then $n$ is just greater than integer.

Permutations and Combinations

Solution:

$^{n-1}C_{3}+ \,{}^{n-1}C_{4} > \,{}^{n}C_{4} >\,{}^{n}C_{3}$
$\Rightarrow \frac{n\,!}{4\,!\left(n-4\right)\,!} > \frac{3\,!}{3\,!\left(n-3\right)\,!}$
$\Rightarrow \frac{1}{4} > \frac{1}{n-3}$
$\Rightarrow n-3 > 4$
$\Rightarrow n > 7$