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Mathematics
Middle term of binomial expansion
Show that the coefficient of middle term of ${{\left( 1+x \right)}^{2n}}$ is equal to the sum of coefficient of the two middle term of ${{\left( 1+x \right)}^{2n-1}}$
Mathematics
Middle term of binomial expansion
Show that the middle term in the expansion of ${{(1+x)}^{2n}}$ is $\dfrac{1\cdot 3\cdot 5\cdot \ldots \cdot (2n-1)}{n!}\cdot {{2}^{n}}{{x}^{n}}$ , where $n\in \mathbb{N}$.
Mathematics
Middle term of binomial expansion
Find the middle term (s) in the expansion of the following ${\left( {\dfrac{a}{b} + \dfrac{a}{b}} \right)^6}$.

Mathematics
Middle term of binomial expansion
The middle term in the expansion of ${{\left( 1-3x+3{{x}^{2}}-{{x}^{3}} \right)}^{2n}}$ is
A.$^{6n}{{C}_{3n}}{{\left( -x \right)}^{3n}}$
B.$^{6n}{{C}_{2n}}{{\left( -x \right)}^{2n+1}}$
C.$^{4n}{{C}_{3n}}{{\left( -x \right)}^{3n}}$
D.$^{6n}{{C}_{3n}}{{\left( -x \right)}^{3n-1}}$
Mathematics
Middle term of binomial expansion
For r = 0, 1, 2,.….10, let \[{{A}_{r}},{{B}_{r}},{{\text{C}}_{r}}\] denote respectively the coefficient of \[{{x}^{r}}\] in the expansions of
\[{{\left( 1+x \right)}^{10}},{{\left( 1+x \right)}^{20}},{{\left( 1+x \right)}^{30}}\] . Then find the value of:
 \[\sum\limits_{r=1}^{10}{{{A}_{r}}\left( {{B}_{r}}{{B}_{10}}-{{C}_{10}}{{A}_{r}} \right)}\]
(a) \[{{B}_{10}}-{{C}_{10}}\]
(b) \[{{A}_{10}}\left( B_{10}^{2}-{{C}_{10}}{{A}_{10}} \right)\]
(c) 0
(d) \[{{C}_{10}}-{{B}_{10}}\]
Mathematics
Middle term of binomial expansion
Show that the middle term in the expansion of ${\left( {1 + x} \right)^n}$is $6{x^2}$ if n = 4.
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