Use the binomial theorem to verify the identity Use the binomial theorem to verify the identity ...

Question:

Use the binomial theorem to verify the identity Use the binomial theorem to verify the identity

{eq}\sum_{k = 0}^n \begin{pmatrix} n \\ k\end{pmatrix} = 2^n {/eq}

Binomial Theorem:


The binomial theorem explains the expansion of powers of a binomial. The theorem says that it is possible to expand the polynomial into the sum of terms, which involves the combinatorics as well which is used to choose coefficients from n.

Answer and Explanation:


The identity to prove is:

{eq}\sum\limits_{k = 0}^n {{}^n{C_k}} = {2^n}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right) {/eq}

The LHS can be proved equal to RHS of the above identity by using binomial theorem which is:

{eq}{\left( {a + b} \right)^n} = {}^n{C_0} \cdot {a^n} \cdot {b^0} + {}^n{C_1} \cdot {a^{n - 1}} \cdot {b^1} + ..... + {}^n{C_n} \cdot {a^0} \cdot {b^n} {/eq}

Taking LHS of (1),

{eq}\begin{align*} \sum\limits_{k = 0}^n {{}^n{C_k}} &= {}^n{C_0} + {}^n{C_1} + ..... + {}^n{C_n}\\ &= {}^n{C_0}{\left( 1 \right)^0}{\left( 1 \right)^n} + {}^n{C_1}{\left( 1 \right)^1}{\left( 1 \right)^{n - 1}} + ..... + {}^n{C_n}{\left( 1 \right)^n}{\left( 1 \right)^0}\\ &= {\left( {1 + 1} \right)^n}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\rm{by}}\,\,{\rm{binomial}}\,\,{\rm{theorem,}}\,\,{\rm{a = 1,b = 1}}} \right)\\ &= {2^n} \end{align*} {/eq}

Since, LHS is equal to RHS,

Therefore, it is proved that {eq}\sum\limits_{k = 0}^n {{}^n{C_k}} = {2^n}. {/eq}


Learn more about this topic:

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The Binomial Theorem: Defining Expressions

from Algebra II: High School

Chapter 12 / Lesson 7
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