Solve the inequation: $\;$ $C^{n}_{6} < C^{n}_{4}$, $\;$ $n \in N$
$C^{n}_{6} < C^{n}_{4}$
i.e. $\;$ $\dfrac{n!}{6! \left(n - 6\right)!} < \dfrac{n!}{4! \left(n - 4\right)!}$
i.e. $\;$ $4! \left(n - 4\right)! < 6! \left(n - 6\right)!$
i.e. $\;$ $4! \left(n - 4\right) \left(n - 5\right) \left(n - 6\right)! < 6 \times 5 \times 4! \left(n - 6\right)!$
i.e. $\;$ $\left(n - 4\right) \left(n - 5\right) < 30$
i.e. $\;$ $n^2 -9n + 20 < 30$
i.e. $\;$ $n^2 - 9n - 10 < 0$
i.e. $\;$ $\left(n - 10\right) \left(n + 1\right) < 0$
i.e. $\;$ $n - 10 < 0$ $\;$ or $\;$ $n + 1 < 0$
i.e. $\;$ $n < 10$ $\;$ or $n < -1$
$\because \;$ $n \in N$ $\implies$ $n = \left\{6, 7, 8, 9\right\}$ $\;$ as per the question, maximum number of $n$ elements taken at a time is $6$.