Question

If $ ^n{C_{r - 1}} = 36,\,{\,^n}{C_r} = 84\,\,and\,{\,^n}{C_{r + 1}} = 126 $ , then r is
 $
  1)\,\,1 \\
  2)\,\,2 \\
  3)\,\,3 \\
  $

Answer 1

Hint: Here in three different terms of combination to find required value of ‘r’ we divide second term by first and then third by second and then using formulas of combination to form two equation in ‘n’ and ‘r’ and then finally solving these equations by method of either substitution or elimination to find value of ‘r’.
 $ ^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $ , $ \dfrac{{^n{C_r}}}{{^n{C_{r - 1}}}} = \dfrac{{n - r + 1}}{r} $

Complete step-by-step answer:
We have given three different terms of combinations. $ ^n{C_{r - 1}} = 36,\,{\,^n}{C_r} = 84\,\,and\,{\,^n}{C_{r + 1}} = 126 $
To find value of ‘r’ we name three different terms as:
  $
  ^n{C_{r - 1}} = 36.................(i) \\
  ^n{C_r} = 84....................(ii) \\
  ^n{C_{r + 1}} = 126...............(iii) \;
  $
Now, on dividing (ii) by (i) we have:
 $
  \dfrac{{^n{C_r}}}{{^n{C_{r - 1}}}} = \dfrac{{84}}{{36}} \\
   \Rightarrow \dfrac{{n - r + 1}}{r} = \dfrac{7}{3} \\
    \\
  on\,\,cross\,\,multiplying\,\,we\,\,have: \\
  3\left( {n - r + 1} \right) = 7r \\
   \Rightarrow 3n - 3r + 3 = 7r \\
   \Rightarrow 3n - 10r + 3 = 0..............(iv) \;
  $
Also, on dividing (iii) by (ii) we have
 $
  \dfrac{{^n{C_{r + 1}}}}{{^n{C_r}}} = \dfrac{{126}}{{84}} \\
   \Rightarrow \dfrac{{n - r}}{{r + 1}} = \dfrac{3}{2} \\
    \\
  on\,\,cross\,\,multiplying\,\,we\,\,have \\
   \Rightarrow 2(n - r) = 3(r + 1) \\
   \Rightarrow 2n - 2r = 3r + 3 \\
   \Rightarrow 2n - 5r - 3 = 0................(v) \;
  $
Solving equation (iv) and (v) by method of substitution. We have:
From equation (v) we have $ n = \dfrac{{3 + 5r}}{2} $ . Substituting this in equation (iv) we have:
\[
  3\left( {\dfrac{{3 + 5r}}{2}} \right) - 10r + 3 = 0 \\
   \Rightarrow \dfrac{{9 + 15r - 20r + 6}}{2} = 0 \\
 \]
\[
   \Rightarrow \dfrac{{15 - 5r}}{2} = 0 \\
   \Rightarrow 15 - 5r = 0 \\
   \Rightarrow 5r = 15 \\
   \Rightarrow r = 3 \;
 \]
Hence, from above we see that the value of r is $ 3 $ .
So, the correct answer is “Option 3”.

Note: The value of ratio of two combination terms can also be found by using combination expansion formula. In this we first use formula to write in terms of factorial then expand big factorial term to cancel small to convert fraction term to an equation. Then solving equations so formed given the value of ‘r’.