(D) The weighted mean is given by $\bar{x} = \frac{\sum_{r=0}^{n} r \cdot {^nC_r}}{\sum_{r=0}^{n} {^nC_r}}$. The numerator is $\sum_{r=0}^{n} r \cdot

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

(D) The weighted mean is given by $\bar{x} = \frac{\sum_{r=0}^{n} r \cdot {^nC_r}}{\sum_{r=0}^{n} {^nC_r}}$. The numerator is $\sum_{r=0}^{n} r \cdot {^nC_r} = \sum_{r=1}^{n} r \cdot \frac{n}{r} \cdot {^{n-1}C_{r-1}} = n \sum_{r=1}^{n} {^{n-1}C_{r-1}} = n \cdot 2^{n-1}$. The denominator is $\sum_{r=0}^{n} {^nC_r} = 2^n$. Therefore, the mean is $\bar{x} = \frac{n \cdot 2^{n-1}}{2^n} = \frac{n}{2}$.

Class 11 Mathematics Binomial Theorem Properties of binomial coefficients

Difficult

The mean of the values $0, 1, 2, \dots, n$ having corresponding weights $^nC_0, ^nC_1, ^nC_2, \dots, ^nC_n$ respectively is

A
$\frac{2^n}{n + 1}$
B
$\frac{2^{n + 1}}{n(n + 1)}$
C
$\frac{n + 1}{2}$
D
$\frac{n}{2}$

Explore More

Browse All

Question Bank

All Subjects

Class 11 Questions

Class 11

Mathematics Questions

Chapter

Binomial Theorem

Subtopic

Properties of binomial coefficients

If $n$ is a positive integer,the value of $(2n+1) ^nC_0 + (2n-1) ^nC_1 + (2n-3) ^nC_2 + \ldots + 1 \cdot ^nC_n$ is

MediumWBJEE 2024

View Solution

The ratio of the coefficients of the terms $x^{n-r}a^r$ and $x^ra^{n-r}$ in the binomial expansion of $(x+a)^n$ is:

Easy

View Solution

The value of the sum $\left({ }^{n} C_{1}\right)^{2}+\left({ }^{n} C_{2}\right)^{2}+\left({ }^{n} C_{3}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}$ is

MediumWBJEE 2014

View Solution

The value of $\sum_{r=0}^{6} \left({}^{6}C_{r} \cdot {}^{6}C_{6-r}\right)$ is equal to:

DifficultJEE MAIN 2021

View Solution

If $\sum_{k=1}^{30} k \left({ }^{30} C _k\right)^2 = \frac{\alpha 60 !}{(30 !)^2}$,then $\alpha$ is equal to

DifficultJEE MAIN 2023

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

The mean of the values $0, 1, 2, \dots, n$ having corresponding weights $^nC_0, ^nC_1, ^nC_2, \dots, ^nC_n$ respectively is

Similar Questions

If $n$ is a positive integer,the value of $(2n+1) ^nC_0 + (2n-1) ^nC_1 + (2n-3) ^nC_2 + \ldots + 1 \cdot ^nC_n$ is

The ratio of the coefficients of the terms $x^{n-r}a^r$ and $x^ra^{n-r}$ in the binomial expansion of $(x+a)^n$ is:

The value of the sum $\left({ }^{n} C_{1}\right)^{2}+\left({ }^{n} C_{2}\right)^{2}+\left({ }^{n} C_{3}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}$ is

The value of $\sum_{r=0}^{6} \left({}^{6}C_{r} \cdot {}^{6}C_{6-r}\right)$ is equal to:

If $\sum_{k=1}^{30} k \left({ }^{30} C _k\right)^2 = \frac{\alpha 60 !}{(30 !)^2}$,then $\alpha$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module