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The no. of positive integers satisfying the inequation $^{n+1}C_{n-2}-^{n+1}C_{n-1}\leq 50$ is ?

$\begin{array}{1 1} 6 \\ 7 \\ 8 \\ 9 \end{array}$

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$^{n+1}C_{n-2}-^{n+1}C_{n-1}\leq 50$
$\Rightarrow\:\large\frac{(n+1)!}{(n-2)!.3!}-\frac{(n+1)!}{(n-1)!.2!}$$\leq 50$
$\Rightarrow\:\large\frac{(n+1)n}{2}$$\bigg[\large\frac{n-1}{3}$$-1\bigg]\leq 50$
$\Rightarrow\:(n+1)n(n-4)\leq 300$
Since $^{n+1}C_{n-2}$ exists for $n\geq 2$,
and for $n=9$, $(n+1)n(n-4)=450\nleq 300$, $n<9$
$i.e., n=2,3,4,5,6,7,8$
$\therefore$ The required no. of positive integers = 7
answered Aug 13, 2013 by rvidyagovindarajan_1
 

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