Question:
If \( x + y = 1 \), then
\[
\sum_{r=0}^{n} r^2 \cdot n C_r x^r y^{n-r} \quad \text{is equal to:}
\]
If \( x + y = 1 \), then
\[
\sum_{r=0}^{n} r^2 \cdot n C_r x^r y^{n-r} \quad \text{is equal to:}
\]
Show Hint
When solving summation problems with binomial coefficients, look for standard identities to simplify the expression.
Updated On: Apr 22, 2026
- \( nxy \)
- \( n x(x + y n) \)
- \( n x (x + y) \)
- None of these
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Use the binomial expansion.
We are asked to evaluate the sum: \[ \sum_{r=0}^{n} r^2 \cdot n C_r x^r y^{n-r} \] Given \( x + y = 1 \), we can expand this sum using the binomial expansion.
Step 2: Apply the identity for binomial sums.
Use the identity for sums involving binomial coefficients and powers: \[ \sum_{r=0}^{n} r^2 \cdot n C_r x^r y^{n-r} = n x(x + y) \] Substituting \( x + y = 1 \): \[ = n x (x + y) = n x \] Thus, the correct answer is \( n x (x + y) \), corresponding to option (C).
We are asked to evaluate the sum: \[ \sum_{r=0}^{n} r^2 \cdot n C_r x^r y^{n-r} \] Given \( x + y = 1 \), we can expand this sum using the binomial expansion.
Step 2: Apply the identity for binomial sums.
Use the identity for sums involving binomial coefficients and powers: \[ \sum_{r=0}^{n} r^2 \cdot n C_r x^r y^{n-r} = n x(x + y) \] Substituting \( x + y = 1 \): \[ = n x (x + y) = n x \] Thus, the correct answer is \( n x (x + y) \), corresponding to option (C).
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Binomial theorem Questions
- The coefficient of $x^{-9}$ in the expansion of $\left(\dfrac{x^{2}}{2} - \dfrac{2}{x}\right)^{9}$ is}
- If the coefficients of the $r$th term and the $(r+1)$th term in the expansion of $(1+x)^{20}$ are in the ratio $1:2$, then $r$ equals
- The first three terms in the expansion of $(1 + ax)^{n}$ $(n \neq 0)$ are $1$, $6x$ and $16x^{2}$. Then the values of $a$ and $n$ are respectively
- The middle term of $\left(x - \frac{1}{x}\right)^6$ is
- The number of terms in the expansion of \( (x + a)^{n} \) is:
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions