the sum 12 10c0 10c1 2 cdot 10c2 2 2 cdot 10c3 2 9 cdot

The sum $$\frac{1}{2}{\,^{10}}{C_0} - {\,^{10}}{C_1} + 2 \cdot {\,^{10}}{C_2} - {2^2} \cdot {\,^{10}}{C_3} + ..... + {2^9} \cdot {\,^{10}}{C_{10}}$$ is equal to

A. $$\frac{1}{2}$$

B. $$0$$

C. $$\frac{1}{2} \cdot {3^{10}}$$

D. None of these

Answer : $$\frac{1}{2}$$

Solution :
Sum $$ = \frac{1}{2}\,\left\{ {^{10}{C_0} - {\,^{10}}{C_1} \cdot 2 + {\,^{10}}{C_2} \cdot {2^2} - {\,^{10}}{C_3} \cdot {2^3} + ..... + {\,^{10}}{C_{10}} \cdot {2^{10}}} \right\}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2}{\left( {1 - 2} \right)^{10}} = \frac{1}{2}.$$

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 $$\frac{1}{2}{\,^{10}}{C_0} - {\,^{10}}{C_1} + 2 \cdot {\,^{10}}{C_2} - {2^2} \cdot {\,^{10}}{C_3} + ..... + {2^9} \cdot {\,^{10}}{C_{10}}$$ is equal to

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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Binomial Theorem

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The sum $$\frac{1}{2}{\,^{10}}{C_0} - {\,^{10}}{C_1} + 2 \cdot {\,^{10}}{C_2} - {2^2} \cdot {\,^{10}}{C_3} + ..... + {2^9} \cdot {\,^{10}}{C_{10}}$$ is equal to

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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Releted MCQ Question on
Algebra >> Binomial Theorem

Practice More Releted MCQ Question on
Binomial Theorem