The ratio of the coefficient of the middle term in the expansion

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 $=\frac{{ }^{20} \mathrm{C}_{10}}{{ }^{19} \mathrm{C}_{9}+{ }^{19} \mathrm{C}_{10}}=\frac{{ }^{20} \mathrm{C}_{10}}{{ }^{20} \mathrm{C}_{10}}=1$

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