Let ${\mathrm{P}}_{\mathrm{n}}$ denotes the number of permutations of n distinct things taken all at a time $x_n{=}^{n+5}{C}_4-\left(\frac{143}{96}\right)\left(\frac{P_{n+5}}{P_{n+3}}\right)$ (where $\text{ nÎ }$ N). The possible value of n for which $x_n$ is negative, can be

$x_n{=}^{n+5}{C}_4-\left(\frac{143}{96}\right)\left(\frac{P_{n+5}}{P_{n+3}}\right)\text{ given }{x}_n<0$

${\text{ Hence }}^{n+5}{C}_4<\frac{143}{96}\cdot \frac{(n+5)!}{(n+2)!}$

$\frac{(n+5)!}{(n+2)!}<\frac{143}{96}\cdot \frac{(n+5)!}{(n+3)(n+2)(n+1)!}$

$\frac{1}{24}<\frac{143}{96}\cdot \frac{1}{(n+3)(n+2)}$

$\Rightarrow 4(n+3)(n+2)<143$

$\Rightarrow 4\left(n^2+5n+6\right)-143<0$

$\Rightarrow 4n^2+20n-119<0$

$\Rightarrow 4n^2+34n-14n-119<0$

$\Rightarrow 2n(2n+17)-7(2n+17)<0$

$\left(2n-7\right)\left(2n+17\right)<0$

$n\in \{1,2,3\}$