First slide

Combinations

Question

If $^{n - 1} C_{3} +^{n - 1} C_{4} >^{n} C_{3},$ then,

Moderate

A

$n \geq 4$
B

$n > 5$
C

$n > 7$
D

none of these

Solution

We have,
$^{n - 1} C_{3} +^{π - 1} C_{4} >^{n} C_{3}$
$=^{n} C_{4} >^{n} C_{3} [∵^{n} C_{r - 1} +^{n} C_{r} =^{n + 1} C_{r}]$
$= \frac{n!}{(n - 4)! 4!} > \frac{n!}{(n - 3)! 3!} \Rightarrow \frac{1}{4} > \frac{1}{n - 3} \Rightarrow n > 7$

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