If {Σlimits_{i = 1}^{20} {left( {frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{Cᵢ} + {}^{20}{C_{i

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7.Binomial Theorem

hard

If ${\sum\limits_{i = 1}^{20} {\left( {\frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3}\, = \frac{k}{{21}}$, then $k$ equals

$400$

$50$

$200$

$100$

(JEE MAIN-2019)

Solution

${\sum\limits_{i = 1}^{20} {\left( {\frac{{{\,^{20}}{C_{I – 1}}}}{{^{20}{C_1} + {\,^{20}}{C_{I – 1}}}}} \right)} ^3}$

Now $\frac{{^{20}{C_{I – 1}}}}{{^{20}{C_1} + {\,^{20}}{C_{I – 1}}}} = \frac{{^{20}{C_{I – 1}}}}{{^{20}{C_1}}} = \frac{1}{{21}}$

Let given sum be $S$, so

$S = \sum\limits_{I = 1}^{20} {\frac{{{{\left( i \right)}^3}}}{{{{21}^3}}}} = \frac{1}{{{{(21)}^3}}}{\left( {\frac{{20.21}}{2}} \right)^2} = \frac{{100}}{{21}}$

Given $S = \frac{k}{{21}} \Rightarrow k = 100$

Standard 11

Mathematics

The coefficient of $x^{91}$ in the series $^{100}{C_1}\,{2^8}.\,{\left( {1\, – \,x} \right)^{99}}\, + {\,^{100}}{C_2}\,{2^7}.\,{\left( {1\, – \,x} \right)^{98}}\, + {\,^{100}}{C_3}\,{2^6}.\,{\left( {1\, – \,x} \right)^{97}}\, + \,….\, + {\,^{100}}{C_9}\,{\left( {1\, – \,x} \right)^{91}}$ is equal to –

normal

The sum of the co-efficients of all odd degree terms in the expansion of ${\left( {x + \sqrt {{x^3} – 1} } \right)^5} + {\left( {x – \sqrt {{x^3} – 1} } \right)^5},\left( {x > 1} \right)$

hard

(JEE MAIN-2018)

The value of $4 \{^nC_1 + 4 . ^nC_2 + 4^2 . ^nC_3 + …… + 4^{n – 1}\}$ is :

normal

The number of terms in the expansion of $(1 +x)^{101} (1 +x^2 – x)^{100}$ in powers of $x$ is

hard

(JEE MAIN-2014)

Let ${\left( {1 + x + {x^2}} \right)^{20}}\left( {2x + 1} \right) = {a_0} + {a_1}{x^1} + {a_2}{x^2} + … + {a_{41}}{x^{41}}$ , then $\frac{{{a_0}}}{1} + \frac{{{a_1}}}{2} + …. + \frac{{{a_{41}}}}{{42}}$ is equal to

normal

If ${\sum\limits_{i = 1}^{20} {\left( {\frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3}\, = \frac{k}{{21}}$, then $k$ equals

Solution

Similar Questions

The sum of the co-efficients of all odd degree terms in the expansion of ${\left( {x + \sqrt {{x^3} – 1} } \right)^5} + {\left( {x – \sqrt {{x^3} – 1} } \right)^5},\left( {x > 1} \right)$

The value of $4 \{^nC_1 + 4 . ^nC_2 + 4^2 . ^nC_3 + …… + 4^{n – 1}\}$ is :

The number of terms in the expansion of $(1 +x)^{101} (1 +x^2 – x)^{100}$ in powers of $x$ is

Let ${\left( {1 + x + {x^2}} \right)^{20}}\left( {2x + 1} \right) = {a_0} + {a_1}{x^1} + {a_2}{x^2} + … + {a_{41}}{x^{41}}$ , then $\frac{{{a_0}}}{1} + \frac{{{a_1}}}{2} + …. + \frac{{{a_{41}}}}{{42}}$ is equal to

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