JEE Main 2021 (Online) 25th July Morning ShiftQuestion 1
The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is $\_\_\_\_\_\_$ .
Binomial Theorem · General Term and Middle Term
Answer$1$
Solution
Concept. The two middle coefficients of $(1+x)^{19}$ add up to the middle coefficient of $(1+x)^{20}$. Formulas used. (1) Pascal's rule: ${}^{n}C_{r-1}+{}^{n}C_r={}^{n+1}C_r$Step 1 - Middle term of $(1+x)^{20}$. $T_{11}$, coefficient ${}^{20}C_{10}$. Step 2 - Middle terms of $(1+x)^{19}$. $T_{10}$ and $T_{11}$, coefficients ${}^{19}C_9$ and ${}^{19}C_{10}$. By (1), their sum is ${}^{20}C_{10}$. Step 3 - Ratio. $\dfrac{{}^{20}C_{10}}{{}^{20}C_{10}}=1$. Answer: 1