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

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Solution

Coefficient of middle term in $(1 + x)^{20} =^{20} C_{10}$
Sum of coefficient of two middle terms in $(1 + x)^{19} =^{19} C_{9} +^{19} C_{10} =^{20} C_{10} (∵^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r})$

Hence, required ratio is $1$

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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

Coefficient of middle term in (1+x)20= 20C10 Sum of coefficient of two middle terms in (1+x)19= 19C9+19C10= 20C10 (∵ nCr+ nCr−1= n+1Cr) Hence, required ratio is 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

Coefficient of middle term in $(1 + x)^{20} =^{20} C_{10}$
Sum of coefficient of two middle terms in $(1 + x)^{19} =^{19} C_{9} +^{19} C_{10} =^{20} C_{10} (∵^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r})$

Hence, required ratio is $1$