Question

English

Gujarati

Hindi

7.Binomial Theorem

hard

The sum of the series $\left( {\begin{array}{{20}{c}}{20}\\0\end{array}} \right) - \left( {\begin{array}{{20}{c}}{20}\\1\end{array}} \right)$$+$$\left( {\begin{array}{{20}{c}}{20}\\2\end{array}} \right) - \left( {\begin{array}{{20}{c}}{20}\\3\end{array}} \right)$$+…..-……+$$\left( {\begin{array}{*{20}{c}}{20}\\{10}\end{array}} \right)$

$0$

$\;\left( {\begin{array}{*{20}{c}}{20}\\{10}\end{array}} \right)$

-$\;\left( {\begin{array}{*{20}{c}}{20}\\{10}\end{array}} \right)$

$\frac{1}{2}\left( {\begin{array}{*{20}{c}}{20}\\{10}\end{array}} \right)$

(AIEEE-2007)

Solution

We know that, $(1+x)^{20}=20 \mathrm{C}_{0}+20 \mathrm{C}_{1} \mathrm{x}+^{20} \mathrm{C}_{2}$

$\mathrm{x}^{2}+\ldots \ldots .20 \mathrm{C}_{10} \mathrm{c}_{10} \mathrm{x}^{10}+\ldots . .2 .20 \mathrm{C}_{20} \mathrm{x}^{20}$

Put $\mathrm{x}=-1 .(0)= ^{20} \mathrm{C}_{0}- ^{20} \mathrm{C}_{1}+ ^{20} \mathrm{C}_{2}- ^{20} \mathrm{C}_{3}+$$\ldots . .+ ^{20} \mathrm{C}_{10}- ^{20} \mathrm{C}_{11} \ldots+^{20} \mathrm{C}_{20}$

$0=2[^{20} \mathrm{C}_{0}- ^{20} \mathrm{C}_{1}+ ^{20} \mathrm{C}_{2}- ^{20} $$\mathrm{C}_{3}+\ldots . .- ^{20} \mathrm{C}_{9} ]+ ^{20} \mathrm{C}_{10}$

$^{20} \mathrm{C}_{10}=2 [^{20} \mathrm{C}_{0}-^{20} \mathrm{C}_{1}$$+ ^{20} \mathrm{C}_{2}-^{20} \mathrm{C}_{3}+\ldots \ldots-^{20} \mathrm{C}_{9}+^{20} \mathrm{C}_{10}]$

$^{20} \mathrm{C}_{0}-^{20} \mathrm{C}_{1}+^{20} \mathrm{C}_{2}-^{20} \mathrm{C}_{3} \cdot \ldots .$ i" $^{2 \mathrm{n}} \mathrm{C}_{10}$

$=\frac{1}{2}^{20} \mathrm{C}_{10}$

Std 11

Mathematics

If the sum of the coefficients in the expansion of ${(1 – 3x + 10{x^2})^n}$ is $a$ and if the sum of the coefficients in the expansion of ${(1 + {x^2})^n}$ is $b$, then

medium

Let $\left( a + bx + cx ^2\right)^{10}=\sum \limits_{ i =0}^{20} p _{ i } x ^{ i }, a , b , c \in N$. If $p _1=20$ and $p _2=210$, then $2( a + b + c )$ is equal to

hard

If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + …. + {C_n}{x^n}$, then the value of ${C_0} + 2{C_1} + 3{C_2} + …. + (n + 1){C_n}$ will be

medium

If the Coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ is $\alpha$, then $|\alpha|$ equals

hard

Let $\left(2 x ^{2}+3 x +4\right)^{10}=\sum \limits_{ r =0}^{20} a _{ r } x ^{ r } \cdot$ Then $\frac{ a _{7}}{ a _{13}}$ is equal to

hard

Solution

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