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7.Binomial Theorem
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Prove that $\sum\limits_{r = 0}^n {{3^r}{\,^n}{C_r} = {4^n}} $
Option A
Option B
Option C
Option D
Solution
By Binomial Theorem,
$\sum\limits_{r = 0}^n {{\,^n}{C_r}{a^{n – r}}{b^r} = {{\left( {a + b} \right)}^n}} $
By putting $b=3$ and $a=1$ in the above equation, we obtain
$\sum\limits_{r = 0}^n {{\,^n}{C_r}{{\left( 1 \right)}^{n – r}}{{\left( 3 \right)}^r} = {{\left( {1 + 3} \right)}^n}} $
$ \Rightarrow \sum\limits_{r = 0}^n {{3^r}{\,^n}{C_r} = {4^n}} $
Hence proved.
Standard 11
Mathematics
