Question:
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________.
Solution:
Coeff. of middle term in $(1+x)^{20}={ }^{20} C_{10}$
\& Sum of Coeff. of two middle terms in
$(1+x)^{19}={ }^{19} C_{9}+{ }^{19} C_{10}$
So required ratio $=\frac{{ }^{20} \mathrm{C}_{10}}{{ }^{19} \mathrm{C}_{9}+{ }^{19} \mathrm{C}_{10}}=\frac{{ }^{20} \mathrm{C}_{10}}{{ }^{20} \mathrm{C}_{10}}=1$