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  4. The inequality n + 1C 6 + nC 4 > n + 2C 5 - nC 5 holds true for all n greater than

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Q. The inequality $\_{}^{n + 1}C _{6} + \_{}^{n}C _{4} > \_{}^{n + 2}C _{5} - \_{}^{n}C _{5}$ holds true for all $n$ greater than

NTA AbhyasNTA Abhyas 2020Permutations and Combinations

Solution:

$\_{}^{n + 1}C_{6}+\_{}^{n}C_{4}+\_{}^{n}C_{5}>\_{}^{n + 2}C_{5}$
$\Rightarrow \_{}^{n + 1}C_{6}+\_{}^{n + 1}C_{5}>\_{}^{n + 2}C_{5}$
$\Rightarrow \_{}^{n + 2}C_{6}>\_{}^{n + 2}C_{5}$
$\Rightarrow \frac{\left(n + 2\right) !}{6 ! \left(n - 4\right) !}>\frac{\left(n + 2\right) !}{5 ! \left(n - 3\right) !}$
$\Rightarrow \frac{1}{6}>\frac{1}{\left(n - 3\right)}$
$\Rightarrow n-3>6$
$\Rightarrow n>9$