Evaluating a Binomial Summation Using the Binomial Theorem

The question asks us to find the value of the summation \(\displaystyle\sum_{r=0}^n\) 2\(^r\) C(n, r), where C(n, r) represents the binomial coefficient \(\binom{n}{r}\).

The given summation can be written as:

\(\displaystyle\sum_{r=0}^n \binom{n}{r} 2^r\)

This summation looks very similar to the expansion formula provided by the Binomial Theorem.

Understanding the Binomial Theorem

The Binomial Theorem states that for any non-negative integer \(n\), the expansion of \((x+y)^n\) is given by:

\((x+y)^n = \displaystyle\sum_{r=0}^n \binom{n}{r} x^{n-r} y^r\)

This formula expands \((x+y)^n\) into a sum of terms, where each term involves a binomial coefficient \(\binom{n}{r}\) and powers of \(x\) and \(y\).

• \(\binom{n}{r}\) is the binomial coefficient, calculated as \(\frac{n!}{r!(n-r)!}\).
• \(r\) is the index of summation, ranging from 0 to \(n\).
• \(x^{n-r}\) is the power of the first term \(x\).
• \(y^r\) is the power of the second term \(y\).

Applying the Binomial Theorem to the Summation

Let's compare the given summation with the Binomial Theorem formula:

• Given Summation: \(\displaystyle\sum_{r=0}^n \binom{n}{r} 2^r\)
• Binomial Theorem Formula: \(\displaystyle\sum_{r=0}^n \binom{n}{r} x^{n-r} y^r\)

We need to find values for \(x\) and \(y\) such that \(\binom{n}{r} x^{n-r} y^r\) matches \(\binom{n}{r} 2^r\) for each term in the sum.

Looking at the term \(\binom{n}{r} 2^r\), we can see that it has the binomial coefficient \(\binom{n}{r}\) and a term \(2^r\). In the Binomial Theorem formula, the term is \(\binom{n}{r} x^{n-r} y^r\).

Comparing the powers of \(y\) and the constant term in the given sum, we can match \(y^r\) with \(2^r\). This suggests \(y = 2\).

Now let's look at the \(x^{n-r}\) part. The given sum \(\binom{n}{r} 2^r\) doesn't explicitly show a term with the power \(n-r\). However, we know that \(1^{n-r} = 1\) for any value of \(n-r\).

So, we can rewrite the given term as \(\binom{n}{r} 1^{n-r} 2^r\).

Now comparing \(\binom{n}{r} 1^{n-r} 2^r\) with \(\binom{n}{r} x^{n-r} y^r\), we can identify:

• \(x^{n-r}\) corresponds to \(1^{n-r}\), so \(x = 1\).
• \(y^r\) corresponds to \(2^r\), so \(y = 2\).

Calculating the Value of the Summation

According to the Binomial Theorem, the sum \(\displaystyle\sum_{r=0}^n \binom{n}{r} x^{n-r} y^r\) is equal to \((x+y)^n\).

Substituting the values \(x = 1\) and \(y = 2\) that we found:

\(\displaystyle\sum_{r=0}^n \binom{n}{r} 1^{n-r} 2^r = (1+2)^n\)

Simplifying the expression on the right side:

\((1+2)^n = 3^n\)

Therefore, the value of the summation \(\displaystyle\sum_{r=0}^n \binom{n}{r} 2^r\) is \(3^n\).

Reviewing the Options

Let's look at the given options:

• Option 1: 2\(^n\)
• Option 2: 3\(^n\)
• Option 3: 2\(^{2n}\)
• Option 4: 3\(^{2n}\)

Our calculated value is \(3^n\), which matches Option 2.

Revision Table: Key Concepts

Additional Information on Binomial Expansions

The Binomial Theorem is a fundamental result in algebra and combinatorics. It has many applications in various fields, including probability, statistics, and calculus.

Some other important binomial expansions and related identities include:

• The sum of binomial coefficients: \(\displaystyle\sum_{r=0}^n \binom{n}{r} = \binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n} = (1+1)^n = 2^n\). (This corresponds to setting \(x=1\) and \(y=1\) in the Binomial Theorem).
• Alternating sum of binomial coefficients: \(\displaystyle\sum_{r=0}^n (-1)^r \binom{n}{r} = \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n \binom{n}{n} = (1-1)^n = 0^n\). (This corresponds to setting \(x=1\) and \(y=-1\). \(0^n\) is 0 for \(n \gt 0\), and 1 for \(n=0\)).
• More generally, for any constant \(c\), \(\displaystyle\sum_{r=0}^n \binom{n}{r} c^r = \displaystyle\sum_{r=0}^n \binom{n}{r} 1^{n-r} c^r = (1+c)^n\). Our problem is a specific case of this identity where \(c=2\).

Recognizing the form of a summation and relating it to the Binomial Theorem is a common technique for evaluating such sums. The key is to identify the \(x\) and \(y\) terms in the general expansion formula that match the terms in the specific summation.