if x y 1 then sumlimitsr 0 n r ncrx ry n r equals 452031008

If $$x + y = 1,$$ then $$\sum\limits_{r = 0}^n {r{\,^n}{C_r}\,{x^r}{y^{n - r}}} $$ equals

A. $$1$$

B. $$n$$

C. $$nx$$

D. $$ny$$

Answer : $$nx$$

Solution :
We have,
$$\eqalign{ & \sum\limits_{r = 0}^n {r{\,^n}{C_r}\,{x^r}{y^{n - r}}} = \sum\limits_{r = 0}^n {r\frac{n}{r}{\,^{n - 1}}{C_{r - 1}}} {x^{r - 1}}{x^1}{y^{n - r}} \cr & = nx\sum\limits_{r = 0}^n {^{n - 1}{C_{r - 1}}{x^{r - 1}}{y^{\left( {n - 1} \right) - \left( {r - 1} \right)}}} \cr & = nx{\left( {x + y} \right)^{n - 1}} = nx\left[ {\because x + y = 1} \right] \cr} $$

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

If $$x + y = 1,$$ then $$\sum\limits_{r = 0}^n {r{\,^n}{C_r}\,{x^r}{y^{n - r}}} $$ equals

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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If $$x + y = 1,$$ then $$\sum\limits_{r = 0}^n {r{\,^n}{C_r}\,{x^r}{y^{n - r}}} $$ equals

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
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