IIT-JEE 1979Question 1
If ${}^{n}C_{r-1}=36$, ${}^{n}C_{r}=84$ and ${}^{n}C_{r+1}=126$, then $r$ is
Permutation and Combination
Solution
Concept. Ratios of neighbouring binomial coefficients give two linear equations in $n,r$. Formulas used. (1) $\frac{{}^nC_r}{{}^nC_{r-1}}=\frac{n-r+1}{r}$Step 1 - First ratio. $\frac{n-r+1}{r}=\frac{84}{36}=\frac73\Rightarrow 3n+3=10r$. Step 2 - Second ratio. $\frac{n-r}{r+1}=\frac{126}{84}=\frac32\Rightarrow 2n=5r+3$. Step 3 - Solve. From Step 1, $n=\frac{10r-3}{3}$. Then $\frac{2(10r-3)}{3}=5r+3\Rightarrow 20r-6=15r+9\Rightarrow r=3$, $n=9$. Step 4 - Check. ${}^9C_2=36$, ${}^9C_3=84$, ${}^9C_4=126$. Answer: C