Evaluate `sum_(r = 0)^(n) 3^r ""^nC_r`

Answer

To evaluate the expression \( \sum_{r=0}^{n} 3^r \binom{n}{r} \), we can utilize the Binomial Theorem. The Binomial Theorem states that:

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

### Step-by-Step Solution:

1. **Identify the terms**: In our case, we can set \( x = 3 \) and \( y = 1 \). This gives us the expression:

\[ (3 + 1)^n = \sum_{r=0}^{n} \binom{n}{r} 3^r 1^{n-r} \]

2. **Simplify the right side**: Since \( 1^{n-r} = 1 \) for any \( r \), we can simplify the right side:

\[ (3 + 1)^n = \sum_{r=0}^{n} \binom{n}{r} 3^r \]

3. **Calculate the left side**: Now, calculate \( (3 + 1)^n \):

\[ (3 + 1)^n = 4^n \]

4. **Conclusion**: Therefore, we have:

\[ \sum_{r=0}^{n} 3^r \binom{n}{r} = 4^n \]

### Final Answer:

\[ \sum_{r=0}^{n} 3^r \binom{n}{r} = 4^n \]