Question

For every integer n > 2 the sum of the expansions \(1 - {}^n{C_1} + {}^n{C_2} + - - - {( - 1)^n}{}.^n{C_n}\) is______
1. n(n - 1)2^{n - 2}
2. 0
3. n 2^{m - 1}
4. \((2n_{c_n})^2\)

Answer 1

Correct Answer - Option 2 : 0

Concept:

As we know, the formula for combinations is

\(^n{C_r} = \frac{{n!}}{{r!\left( {n - r} \right)!}}\)

Note:

 \(^n{C_n}=1\)

\(^n{C_r} = {}^n{C_{n - r}}\)

Calculation:

Given, 

n > 2

Let's take n = 3 and put in the series,

⇒ \(1 - {}^n{C_1} + {}^n{C_2} + - - - {( - 1)^n}{}.^n{C_n}\) 

⇒ \(1 - {}^3{C_1} + {}^3{C_2} + - - - {( - 1)^3}.{}^3{C_3}\)

Now Expanding and solving, we get

\(\begin{array}{l} \Rightarrow 1 - \frac{{3!}}{{1!\left( {3 - 1} \right)!}} + \frac{{3!}}{{2!\left( {3 - 2} \right)!}} - 1\\ \Rightarrow 1 - \frac{{3!}}{{1!2!}} + \frac{{3!}}{{2!1!}} - 1 = 0 \end{array}\)

Similarly, we can check for other n values, the sum of the expansions \( - {}^n{C_1} + {}^n{C_2} + - - - {( - 1)^n}{}^n{C_n}\) will come zero for all n > 2.