MCQ
If $^n{C_4},{\,^n}{C_5},$ and ${\,^n}{C_6},$ are in $A.P.,$ then $n$ can be 
  • A
    $9$
  • $14$
  • C
    $11$
  • D
    $12$

Answer

Correct option: B.
$14$
b
$2.{\,^n}{C_5}{ = ^n}{C_4}{ + ^n}{C_6}$

$2.\frac{{\left| n \right.}}{{\left| {5\left| {n - 5} \right.} \right.}} = \frac{{\left| n \right.}}{{\left| {4\left| {n - 4} \right.} \right.}} + \frac{{\left| n \right.}}{{\left| {6\left| {n - 6} \right.} \right.}}$

$\frac{2}{5}.\frac{1}{{n - 5}} = \frac{1}{{\left( {n - 4} \right)\left( {n - 5} \right)}} + \frac{1}{{30}}$

$n=14$ satisfying equation.

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