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

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

WBJEEWBJEE 2010Permutations and Combinations

Solution:

${ }^{n-1} C_{3}+{ }^{n-1} C_{4}>{ }^{n} C_{3}$
$\Rightarrow { }^{n} C_{4}>{ }^{n} C_{3}$
$\left[\because{ }^{n} C_{r}+{ }^{n} C_{r+1}={ }^{n+1} C_{r+1}\right]$
$\Rightarrow \frac{n !}{4 !(n-4) !}>\frac{n !}{3 !(n-3) !}$
$\Rightarrow \frac{1}{4}>\frac{1}{(n-3)}$
$\Rightarrow n-3 > 4$
$ \Rightarrow n > 7$