C_1+2 C_2+3 C_3+ +n C_n =? - Vidyadip

MCQ

$\left\{C_1+2 C_2+3 C_3+\ldots+n C_n\right\}=?$

A
$( n -1) \cdot 2^{ n }$
B
$n \cdot 2^n$
C
$( n +1) \cdot 2^n$
✓
$n \cdot 2^{n-1}$

✓

Answer

Correct option: D.

$n \cdot 2^{n-1}$

$C _1+2 C _2+3 C _3+\ldots+n C _{ n }$
$= n +2 \cdot \frac{ n ( n -1)}{2}+3 \cdot \frac{n(n-1)(n-2)}{3!}+\ldots+n$
$= n \cdot\left[1+( n -1) \frac{( n -1)( n -2)}{2!}+\ldots+1\right]$
$= n \cdot\left[{ }^{( n -1)} C _0+{ }^{( n -1)} C _1+{ }^{( n -1)} C _2+\ldots+{ }^{( n -1)} C _{ n -1}\right]$
$= n \cdot(1+1)^{ n -1}= n \cdot 2 ^{ n -1}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Model Paper 5→Model Paper 5 — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Period of $\sin \theta \cos \theta $ is

The total number of seven digit numbers the sum of whose digits is even is

The number of distinct real roots of the equation $|\mathrm{x}+1||\mathrm{x}+3|-4|\mathrm{x}+2|+5=0$, is ...........

Let $3,6,9,12, \ldots$ upto $78$ terms and $5,9,13,17, \ldots$ upto $59$ terms be two series. Then, the sum of the terms common to both the series is equal to

Consider the following frequency distribution:

Class:	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$
Freq:	$\alpha$	$110$	$54$	$30$	$\beta$

If the sum of all frequencies is $584$ and median is $45$ , then $|\alpha-\beta|$ is equal to $.....$

If $A, B$ and $C$ are any three sets, then $\text{A}\times (\text{B}\cup\text{C})$ is equal to.

The number of four digit numbers that can be made with the digits $1,2,3,4$ and $5$ in which at least two digits are identical is

Let $x_1, x_2, ..., x_n$ be $n$ observations. Let $y_i = ax_i + by_i + b$ for $i = 1, 2, 3, ..., n,$ where $a$ and $b$ are constants. If the mean of $x_i's$ is $48$ and their standard deviation is $12,$ the mean of $y_i's$ $55$ and standard deviation of $y_i's$ is $15,$ the values of $a$ and $b$ are:

If $x^{n-1}$ is divisible by $x - k,$ then the least positive integral value of $k$ is:

The sum of all two digit numbers which, when divided by $4$, yield unity as a remainder is