the sum of the series 20c0 20c1 20c2 20c3 20c10 is 506002096

The sum of the series $$^{20}{C_0} - {\,^{20}}{C_1} + {\,^{20}}{C_2} - {\,^{20}}{C_3} + ..... - ..... + {\,^{20}}{C_{10}}$$ is

A. $$0$$

B. $$^{20}{C_{10}}$$

C. $$ - ^{20}{C_{10}}$$

D. $${\frac{1}{2}^{20}}{C_{10}}$$

Answer : $${\frac{1}{2}^{20}}{C_{10}}$$

Solution :
We know that, $${\left( {1 + x} \right)^{20}} = {\,^{20}}{C_0} - {\,^{20}}{C_1}x + {\,^{20}}{C_2}{x^2} + ..... + {\,^{20}}{C_{10}}{x^{10}} + .....{\,^{20}}{C_{20}}{x^{20}}$$
$$\eqalign{ & {\text{Put }}x = - 1,\,\,\left( 0 \right) = {\,^{20}}{C_0} - {\,^{20}}{C_1} + {\,^{20}}{C_2} - {\,^{20}}{C_3} + ..... + {\,^{20}}{C_{10}} - {\,^{20}}{C_{11}}..... + {\,^{20}}{C_{20}} \cr & \Rightarrow \,\,0 = 2\left[ {^{20}{C_0} - {\,^{20}}{C_1} + {\,^{20}}{C_2} - {\,^{20}}{C_3} + ..... - {\,^{20}}{C_9} + {\,^{20}}{C_{10}}} \right] \cr & \Rightarrow \,{\,^{20}}{C_{10}} = 2\left[ {^{20}{C_0} - {\,^{20}}{C_1} + {\,^{20}}{C_2} - {\,^{20}}{C_3} + ..... - {\,^{20}}{C_9} + {\,^{20}}{C_{10}}} \right] \cr & \Rightarrow \,{\,^{20}}{C_0} - {\,^{20}}{C_1} + {\,^{20}}{C_2} - {\,^{20}}{C_3} + ..... + {\,^{20}}{C_{10}} = \frac{1}{2}{\,^{20}}{C_{10}} \cr} $$

Releted MCQ Question on
Algebra >> Binomial Theorem

Releted Question 1

Given positive integers $$r > 1, n > 2$$ and that the co - efficient of $${\left( {3r} \right)^{th}}\,{\text{and }}{\left( {r + 2} \right)^{th}}$$ terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$ are equal. Then

A. $$n = 2r$$

B. $$n = 2r + 1$$

C. $$n = 3r$$

D. none of these

View Answer

Releted Question 2

The co-efficient of $${x^4}$$ in $${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$$ is

A. $$\frac{{405}}{{256}}$$

B. $$\frac{{504}}{{259}}$$

C. $$\frac{{450}}{{263}}$$

D. none of these

View Answer

Releted Question 3

The expression $${\left( {x + {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5}$$ is a polynomial of degree

A. 5

B. 6

C. 7

D. 8

View Answer

Releted Question 4

If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$ the co-efficients of $$x$$ and $${x^2}$$ are $$3$$ and $$- 6\,$$ respectively, then $$m$$ is

A. 6

B. 9

C. 12

D. 24

View Answer

The sum of the series $$^{20}{C_0} - {\,^{20}}{C_1} + {\,^{20}}{C_2} - {\,^{20}}{C_3} + ..... - ..... + {\,^{20}}{C_{10}}$$ is

Releted MCQ Question on
Algebra >> Binomial Theorem

Given positive integers $$r > 1, n > 2$$ and that the co - efficient of $${\left( {3r} \right)^{th}}\,{\text{and }}{\left( {r + 2} \right)^{th}}$$ terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$ are equal. Then

The co-efficient of $${x^4}$$ in $${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$$ is

The expression $${\left( {x + {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5}$$ is a polynomial of degree

If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$ the co-efficients of $$x$$ and $${x^2}$$ are $$3$$ and $$- 6\,$$ respectively, then $$m$$ is

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The sum of the series $$^{20}{C_0} - {\,^{20}}{C_1} + {\,^{20}}{C_2} - {\,^{20}}{C_3} + ..... - ..... + {\,^{20}}{C_{10}}$$ is

Releted MCQ Question on Algebra >> Binomial Theorem

Given positive integers $$r > 1, n > 2$$ and that the co - efficient of $${\left( {3r} \right)^{th}}\,{\text{and }}{\left( {r + 2} \right)^{th}}$$ terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$ are equal. Then

The co-efficient of $${x^4}$$ in $${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$$ is

The expression $${\left( {x + {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5}$$ is a polynomial of degree

If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$ the co-efficients of $$x$$ and $${x^2}$$ are $$3$$ and $$- 6\,$$ respectively, then $$m$$ is

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