( * 20 c n 0 ) + 2 , ( * 20 c n 1 ) + 2^2 ( * 20 c n 2 ) + ..... … - Vidyadip

MCQ

$\left( {\begin{array}{*{20}{c}}n\\0\end{array}} \right) + 2\,\left( {\begin{array}{*{20}{c}}n\\1\end{array}} \right) + {2^2}\left( {\begin{array}{*{20}{c}}n\\2\end{array}} \right) + ..... + {2^n}\left( {\begin{array}{*{20}{c}}n\\n\end{array}} \right)$ is equal to

A
${2^n}$
B
$0$
✓
${3^n}$
D
None of these

✓

Answer

Correct option: C.

${3^n}$

c
(c) ${(1 + x)^n} = {}^n{C_0} + x.{}^n{C_1} + {x^2}.{}^n{C_2} + .... + {x^n}.{}^n{C_n}$

Put $x = 2$

==> ${3^n} = {}^n{C_0} + 2.{}^n{C_1} + {2^2}.{}^n{C_2} + {2^3}.{}^n{C_3} + .... + {2^n}{.^n}{C_n}$.

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