JEE Main 2020 (Online) 8th January Morning ShiftQuestion 1
If $a,b$ and $c$ are the greatest values of ${}^{19}C_p$, ${}^{20}C_q$ and ${}^{21}C_r$ respectively, then :
Binomial Theorem · Greatest Term and Greatest Coefficient
Solution
Concept. The greatest coefficients are the middle ones; relate them by the ratio formula. Formulas used. (1) Greatest ${}^{n}C_r$: $r=\frac n2$ for even $n$, $r=\frac{n\pm1}{2}$ for odd $n$(2) ${}^{n+1}C_{r+1}=\dfrac{n+1}{r+1}\,{}^{n}C_r$ and ${}^{n+1}C_r=\dfrac{n+1}{n+1-r}\,{}^{n}C_r$Step 1 - The three values. By (1): $a={}^{19}C_9$, $b={}^{20}C_{10}$, $c={}^{21}C_{10}$. Step 2 - $b$ in terms of $a$. By (2), $b=\dfrac{20}{10}\,{}^{19}C_9=2a$. Step 3 - $c$ in terms of $b$. By (2), $c=\dfrac{21}{11}\,{}^{20}C_{10}=\dfrac{21}{11}\cdot2a=\dfrac{42a}{11}$. Step 4 - Compare. $\dfrac a{11}=\dfrac{2a}{22}=\dfrac{42a/11}{42}$, so $\dfrac a{11}=\dfrac b{22}=\dfrac c{42}$. Answer: (D)