Ratio of coefficient of middle term in expansion of (1+x)^{20} and sum of coefficients of two middle

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7.Binomial Theorem

medium

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 $....$

$5$

$4$

$1$

$11$

(JEE MAIN-2021)

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

Standard 11

Mathematics

The sum of the rational terms in the binomial expansion of ${\left( {{2^{\frac{1}{2}}} + {3^{\frac{1}{5}}}} \right)^{10}}$ is

hard

(JEE MAIN-2013)

If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{ax}}{ }^2+\frac{1}{2 \mathrm{x}^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :

medium

(JEE MAIN-2024)

The coefficient of $x^{37}$ in the expansion of $(1-x)^{30} \, (1 + x + x^2)^{29}$ is :

normal

If coefficient of ${(2r + 3)^{th}}$ and ${(r – 1)^{th}}$ terms in the expansion of ${(1 + x)^{15}}$ are equal, then value of r is

medium

If the constant term in the expansion of $\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$ is $p$, then $108$ p is equal to………………..

hard

(JEE MAIN-2024)

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

Similar Questions

The sum of the rational terms in the binomial expansion of ${\left( {{2^{\frac{1}{2}}} + {3^{\frac{1}{5}}}} \right)^{10}}$ is

If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{ax}}{ }^2+\frac{1}{2 \mathrm{x}^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :

The coefficient of $x^{37}$ in the expansion of $(1-x)^{30} \, (1 + x + x^2)^{29}$ is :

If coefficient of ${(2r + 3)^{th}}$ and ${(r – 1)^{th}}$ terms in the expansion of ${(1 + x)^{15}}$ are equal, then value of r is

If the constant term in the expansion of $\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$ is $p$, then $108$ p is equal to………………..

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