question_answer

The value of ?n? for which \[^{n-1}{{C}_{4}}{{-}^{n-1}}{{C}_{3}}-\frac{5}{4}{{.}^{n-1}}{{P}_{2}}<0,\] Where \[n\in N\]

A) \[\{5,6,7,8,9,10\}\]

B) \[\{1,2,3,4,5,6,7,8,9,10\}\]

C) \[\{1,4,5,6,7,8,9,10\}\]

D) \[(-\infty ,2)\cup (3,11)\]

Correct Answer: A

Solution :

[a] we have, \[^{n-1}{{C}_{4}}{{-}^{n-1}}{{C}_{3}}-\frac{5}{4}{{,}^{n-2}}{{P}_{2}}<0\]

\[\Rightarrow \frac{(n-1)(n-2)(n-3)(n-4)}{4!}-\frac{(n-1)(n-2)(n-3)}{3!}\]

\[-\frac{5}{4}(n-2)(n-3)<0\]

\[\Rightarrow (n-2)(n-3)(n-11)(n+2)<0\]

\[\Rightarrow (n-2)(n-3)(n-11)<0\]

\[[\because n+2>0forn\in N]\]

\[\Rightarrow n\in (-\infty ,2)\cup (3,11)\]

\[\Rightarrow n\in (0,2)\cup (3,11)\]

\[\Rightarrow n=1,4,5,6,7,8,9,10\]

But \[^{n-1}{{C}_{4}}\] and \[^{n-2}{{P}_{2}}\] both are meaningful for \[n\ge 5.\]

Hence,

\[n=5,6,7,8,9,10.\]