{n^n}{left( {frac{{n + 1}}{2}} right)^{2n}} is

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7.Binomial Theorem

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${n^n}{\left( {\frac{{n + 1}}{2}} \right)^{2n}}$ is

A

Less than ${\left( {\frac{{n + 1}}{2}} \right)^3}$

B

Greater than ${\left( {\frac{{n + 1}}{2}} \right)^3}$

C

Greater than ${(n!)^3}$

D

$(b)$ and $(c)$ both

Solution

(b) $y = {n^n}{\left( {\frac{{n + 1}}{2}} \right)^{2n}}$

Put $n = 2$, $y = {2^2}{\left( {\frac{3}{2}} \right)^4} = 4\,.\,\frac{{81}}{{8 \times 2}} = \frac{{81}}{4}\tilde – 20$

Option $(a) = {\left( {\frac{{n + 1}}{2}} \right)^3} = \frac{{27}}{8} < y$

Option $(b) = {\left( {\frac{{n + 1}}{2}} \right)^3} = \frac{{27}}{8} < y$

Option $(c) = {(2!)^3} = 8 < y$

Standard 11

Mathematics

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