The value of^n C_1 _ r = 0 ^1 ^1 C_r + ^n C_2 ( _ r = 0 ^2 ^2 C_r… - Vidyadip

MCQ

The value of$^n{C_1}\sum\limits_{r = 0}^1 {^1{C_r}} { + ^n}{C_2}\left( {\sum\limits_{r = 0}^2 {^2{C_r}} } \right){ + ^n}{C_3}\left( {\sum\limits_{r = 0}^3 {^3{C_r}} } \right) + ......{ + ^n}{C_n}\left( {\sum\limits_{r = 0}^n {^n{C_r}} } \right)$ is equal to

A
$2^n$
B
$3^n$
✓
$(3^n-1)$
D
$(3^n+1)$

✓

Answer

Correct option: C.

$(3^n-1)$

c
$^n{C_1} \cdot {2^1} + {{\mkern 1mu} ^n}{C_2} \cdot {2^2} + {{\mkern 1mu} ^n}{C_3} \cdot {2^3} + \ldots . + {{\mkern 1mu} ^n}{C_n} \cdot {2^n}$

${(1 + 2)^n} = {{\mkern 1mu} ^n}{C_0} + {{\mkern 1mu} ^n}{C_1} \cdot {2^1} + {{\mkern 1mu} ^n}{C_2} \cdot {2^2} + \ldots . + {{\mkern 1mu} ^n}{C_n} \cdot {2^n}$

$\left( {{3^n} - 1} \right) = {\,^n}{C_1} \cdot {2^1} + {\,^n}{C_2} \cdot {2^2} + \ldots \ldots + {\,^n}{C_n} \cdot {2^n}$

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