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**Hint:**We got the ${n^{th}}$ term as \[{\left( { - 1} \right)^{n - 1}}\left( {n + 2} \right){C_n}\] and the given series is

\[\dfrac{{i = n}}{{\dfrac{\sum }{{i = 2}}}}{\left( { - 1} \right)^{n - 1}}{\left( {i + 2} \right)^n}Ci\]

Divide the express into 2 and relate it with binomial expansion’s coefficient where $x = - 1$ for ${\left( {1 + x} \right)^n}$

**Complete step-by-step answer:**Let’s begin with given expansion. It is

\[ \Rightarrow {3^n}{C_1} - {4^ n }{C_2} + {5^n}{C_3} - ........ + {\left( { - 1} \right)^{n - 1}}{\left( {n + 2} \right)^n}{C_ n}\]

So we can take ${n^{th}}$ term as

$ \Rightarrow {\left( { - 1} \right)^{r - 1}}\left( {r + 2} \right){\;^n}{C_r}$

Hence, series can also be written as

$\dfrac{n}{{\dfrac{\sum }{{r = 1}}}}{\left( { - 1} \right)^{n - 1}}\left( {r + 2} \right){\;^n}{C_r}$ $\dfrac{{\dfrac{n}{\sum }}}{{r = 1}}\left[ {{{\left( { - 1} \right)}^{r - 1}}r{\;^n}{C_r} + {{\left( { - 1} \right)}^{r - 1}}2.\;2{\;^n}{C_r}} \right]$

So we can divide the expression into Z. where $\dfrac{{\dfrac{n}{\sum }}}{{r = 1}}{\left( { - 1} \right)^{r - 1}}r.{\;^n}{C_r}\;\;\;\& \,2.\dfrac{{\dfrac{n}{\sum }}}{{r = 1}}{\;^n}{C_r}$

First, I would like to solve $\dfrac{{\dfrac{n}{\sum }}}{{i = 1}}{\left( { - 1} \right)^r}\;r{\;^n}{c_r}$. By observing the equation formation. Each of the values having term r as a multiple is quite different from a regular expression. It can be obtained on a regular basis , if the variable is differentiator.

As we know that binomial expansion of

${\left( {1 + x} \right)^ n }{ = ^n}{C_0}{ + ^n}{C_1}x{ + ^n}{C_2}{x^2}{ + ^n}{C_3}{x^3} + ......{ + ^n}{C_n}{x^n}$

If we differentiate both the side, we get $w.r.t\;x$

\[ \Rightarrow n{\left( {1 + x} \right)^{n - 1}} = 0{ + ^n}{C_1}{ + ^n}{C_2}.2x{ + ^n}{C_3}.3{x^{^2}} + {.....^n}{C_n} * n.{x^{n - 1}}\]

\[ \Rightarrow n{\left( {1 + x} \right)^{n - 1}}{ = ^n}{C_1} + {2^n}.{C_2}x + {3.^n}{C_3}.{x^2} + ...... - 1n{.^n}{C_n}.{x^{n - 1}}\]

To obtain the relation with $\left( { - 1} \right)$ in each term we can use $x = - 1.$ so we get.

$ \Rightarrow n{\left( {1 - 1} \right)^{n - 1}} = 0{ + ^n}{C_1} + {2.^n}{C_2}\left( { - 1} \right) + {3.^n}{C_3}{\left( { - 1} \right)^2} + ...... + n{.^n}{C_n}\left( { - 1} \right)$

$ \Rightarrow 0 = 0{ + ^n}{C_1} - {2.^C}{n_2} + {3.^n}{C_3} - {4.^n}{C_4}......$

Hence. We got

$\dfrac{{\dfrac{n}{\sum }}}{{r = 1}}{\left( { - 1} \right)^{r = 1}}r{.^n}{C_r} = 0$ ①

Now, let’s compute

\[\dfrac{{\dfrac{n}{\sum }}}{{r = 1}}{\left( { - 1} \right)^x}{C_r},\]

Which can be computed from Coefficient of ${\left( {1 + x} \right)^ n }$ Binomial expansion will be

${\left( {1 + x} \right)^n}{ = ^n}{C_0}{ + ^n}{C_1}x{ + ^n}{C_2}{x^2} + ......{ + ^n}{C_n}{x^n}$

If if need the expression in the form

\[\left[ {^n{C_1}{ - ^n}{C_2}{ + ^n}{C_3}{ - ^n}{C_4} + .........} \right]\]

We need to put a Value of $x = - 1$ . Therefore the equation will give

$ \Rightarrow {\left( {1 - 1} \right)^n}{ = ^n}{C_0}{ + ^n}{C_1}\left( { - 1} \right){ + ^n}{C_2}{\left( { - 1} \right)^2} + ........{ + ^n}{C_n}{\left( { - 1} \right)^n}$

$ \Rightarrow 0{ = ^n}{C_0}{ - ^n}{C_1}{ + ^n}{C_2}{ - ^n}{C_3} + .........{ + ^n}{C_n}{\left( { - 1} \right)^n}$

If we compare, the equation resulted in

$ \Rightarrow 0{ = ^n}{C_0} - \left[ {^n{C_1}{ - ^n}{C_2}{ + ^n}{C_3}{ - ^n}{C_4} + ...... + {{\left( { - 1} \right)}^{n - 1}}^n{C_n}} \right]$

${ = ^n}{C_0} - \dfrac{{\dfrac{n}{\sum }}}{{r = 1}}{\left( { - 1} \right)^{r - 1}}^n{C_r}$

Therefore, $\dfrac{{\dfrac{n}{\sum }}}{{r = 1}}{\left( { - 1} \right)^{n - 1}}^n{C_r}{ = ^n}{C_0} = 1$

We required $2 \times \dfrac{{\dfrac{n}{\sum }}}{{r = 1}}{\left( { - 1} \right)^{n - 1}}^n{C_r} = 2 \times \left( 1 \right) = 2$ (2)

Hence we got both the value. So the equation given

$ \Rightarrow \dfrac{{\dfrac{n}{\sum }}}{{r = 1}}{\left( { - 1} \right)^{r - 1}}{\left( {r + 1} \right)^n}{C_r} + 2\dfrac{{\dfrac{n}{\sum }}}{{r = 1}}{\left( { - 1} \right)^{r - 1}}^n{C_r}$

Using (1) and (2) we got

$ \Rightarrow 0 + 2\left( 1 \right)$

$ = 2$

**Hence, option B is the correct answer.**

**Note:**If we get into any binomial form of expression. It will be anyhow, the form of Binomial Expression for any short Expression. Binomial Properties are used to shorten the calculation like.

${ \Rightarrow ^n}{C_r}{ + ^n}{C_{r - 1}}{ = ^{n + 1}}{C_r}$

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