Haryana Police Constable Model Paper 4

CONCEPT:
Binomial Theorem:
According to the theorem, it is possible to expand any non-negative power of x+ y into a sum of the form:
${\left(\mathrm{x}+\mathrm{y}\right)}^{\mathrm{n}}={}^{\mathrm{n}}{\mathrm{C}}_0{\mathrm{x}}^{\mathrm{n}}{\mathrm{y}}^0+{}^{\mathrm{n}}{\mathrm{C}}_1{\mathrm{x}}^{\mathrm{n}-1}{\mathrm{y}}^1+{}^{\mathrm{n}}{\mathrm{C}}_2{\mathrm{x}}^{\mathrm{n}-2}{\mathrm{y}}^2+{}^{\mathrm{n}}{\mathrm{C}}_3{\mathrm{x}}^{\mathrm{n}-3}{\mathrm{y}}^3+......+{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{\mathit{x}}^0{\mathrm{y}}^{\mathrm{n}}$ where $\mathrm{n}\ge 0$ is an integer.
CALCULATION:
${\left(1+\mathrm{x}\right)}^{50}=\left(1+{}^{50}{\mathrm{C}}_1{\mathrm{x}}^1+{}^{50}{\mathrm{C}}_2{\mathrm{x}}^2+{}^{50}{\mathrm{C}}_3{\mathrm{x}}^3+{}^{50}{\mathrm{C}}_4{\mathrm{x}}^4+......+{}^{50}{\mathrm{C}}_{50}{\mathrm{x}}^{50}\right)$ ......equation (1)
The above equations is written with the help of “BINOMIAL THEOREM”
We need to find the sum of $\mathrm{left}\left({}^{50}{\mathrm{C}}_1+{}^{50}{\mathrm{C}}_3+{}^{50}{\mathrm{C}}_5+.......+{}^{50}{\mathrm{C}}_{49}\right)$ , as they are the coefficients of odd powers of $\mathrm{x}$ .
In the equation (1), substitute x = 1
$\Rightarrow {\left(1+1\right)}^{50}=1+{}^{50}{\mathrm{C}}_1+{}^{50}{\mathrm{C}}_2+{}^{50}{\mathrm{C}}_3+{}^{50}{\mathrm{C}}_4+......+{}^{50}{\mathrm{C}}_{50}$ $\Rightarrow 2^{50}=1+{}^{50}{\mathrm{C}}_1+{}^{50}{\mathrm{C}}_2+{}^{50}{\mathrm{C}}_3+{}^{50}{\mathrm{C}}_4+......+{}^{50}{\mathrm{C}}_{50}......$ equation (2)
In the equation(1), substitute x = (-1)
$\Rightarrow {\left(1-1\right)}^{50}=1-{}^{50}{\mathrm{C}}_1+{}^{50}{\mathrm{C}}_2-{}^{50}{\mathrm{C}}_3+{}^{50}{\mathrm{C}}_4\mathrm{+̣}.....+{}^{50}{\mathrm{C}}_{50}$ $\Rightarrow 0=1-{}^{50}{\mathrm{C}}_1+{}^{50}{\mathrm{C}}_2-{}^{50}{\mathit{C}}_3+{}^{50}{\mathrm{C}}_4+......+{}^{50}{\mathrm{C}}_{50}......$equation (3)
By subtracting equation (3) from equation (2),
$\Rightarrow 2^{50}=2\times ({}^{50}{\mathrm{C}}_1+{}^{50}{\mathrm{C}}_3+{}^{50}{\mathrm{C}}_5+{}^{50}{\mathrm{C}}_7+......+{}^{50}{\mathrm{C}}_{49}$ $\Rightarrow {}^{50}{\mathrm{C}}_1+{}^{50}{\mathrm{C}}_3+{}^{50}{\mathrm{C}}_5+{}^{50}{\mathrm{C}}_7+.......+{}^{50}{\mathrm{C}}_{49}=2^{49}$
The sum of the coefficients of odd powers of $\mathrm{x}=2^{49}$