Permutations and Combinations
Properties of Combinations
IIT-JEE 1979
Grade 11
Question:
If $^nC_{r-1} = 36$, $^nC_r = 84$ and $^nC_{r+1} = 126$, then $r$ equals:
(1) $1$
(2) $2$
(3) $3$
(4) $4$
Step-by-Step Solution
Key Concept: Form ratios: $\frac{^nC_r}{^nC_{r-1}} = \frac{n-r+1}{r} = \frac{84}{36} = \frac{7}{3}$ gives $3n-10r+3 = 0$. Similarly $\frac{^nC_{r+1}}{^nC_r} = \frac{n-r}{r+1} = \frac{126}{84} = \frac{3}{2}$ gives $2n - 5r - 3 = 0$. Solve simultaneously to get $r = 3$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)