the coefficient of x 3 in the expansion of 1 x x 2 5 is

The coefficient of $$x^3$$ in the expansion of $${\left( {1 - x + {x^2}} \right)^5}$$ is

A. $$10$$

B. $$- 20$$

C. $$- 50$$

D. $$- 30$$

Answer : $$- 30$$

Solution :
$$\eqalign{ & {\left( {1 - x + {x^2}} \right)^5} = {\left\{ {1 + x\left( {x - 1} \right)} \right\}^5} \cr & {\left( {1 - x + {x^2}} \right)^5} = {\,^5}{C_0} + {\,^5}{C_1}x\left( {x - 1} \right) + {\,^5}{C_2}{x^2}{\left( {x - 1} \right)^2} + {\,^5}{C_3}{x^3}{\left( {x - 1} \right)^3} + ..... \cr} $$
∴ the co-efficient of $${x^3} = - 2 \cdot {\,^5}{C_2} - {\,^5}{C_3}.$$

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 coefficient of $$x^3$$ in the expansion of $${\left( {1 - x + {x^2}} \right)^5}$$ 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 coefficient of $$x^3$$ in the expansion of $${\left( {1 - x + {x^2}} \right)^5}$$ 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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