`(1+2x+3x^(2))^(15) = sum_(r=0)^(30) a_(r)x^(r)` then digit at the unit place of `a_(0) + a_(1) + a_(30)` is

Explanation

To solve the problem, we need to find the digit at the unit place of \( a_0 + a_1 + a_{30} \) from the expression \( (1 + 2x + 3x^2)^{15} = \sum_{r=0}^{30} a_r x^r \).

### Step-by-step Solution:

1. **Identify \( a_0 \)**:
- \( a_0 \) is the constant term in the expansion of \( (1 + 2x + 3x^2)^{15} \).
- When \( x = 0 \), the expression simplifies to \( (1 + 0 + 0)^{15} = 1^{15} = 1 \).
- Thus, \( a_0 = 1 \).

**Hint**: To find the constant term, substitute \( x = 0 \) in the expression.

2. **Identify \( a_1 \)**:
- \( a_1 \) is the coefficient of \( x \) in the expansion.
- To find \( a_1 \), we differentiate the expression and then substitute \( x = 0 \).
- Differentiate \( (1 + 2x + 3x^2)^{15} \):
\[ \frac{d}{dx}[(1 + 2x + 3x^2)^{15}] = 15(1 + 2x + 3x^2)^{14} \cdot (2 + 6x) \]
- Now, substitute \( x = 0 \):
\[ 15(1 + 0 + 0)^{14} \cdot (2 + 0) = 15 \cdot 1^{14} \cdot 2 = 30 \]
- Thus, \( a_1 = 30 \).

**Hint**: Differentiate the expression and evaluate at \( x = 0 \) to find the coefficient of \( x \).

3. **Identify \( a_{30} \)**:
- \( a_{30} \) is the coefficient of \( x^{30} \).
- The maximum degree of \( x \) in the expansion is \( 30 \), which occurs when we select \( 3x^2 \) from all \( 15 \) factors.
- Therefore, \( a_{30} = 3^{15} \).

**Hint**: To find the coefficient of the highest degree, consider the contributions from the terms that can yield that degree.

4. **Calculate \( a_0 + a_1 + a_{30} \)**:
- Now we have:
\[ a_0 = 1, \quad a_1 = 30, \quad a_{30} = 3^{15} \]
- We need to find \( 3^{15} \) and its unit digit:
- The unit digit of powers of \( 3 \) follows the pattern: \( 3, 9, 7, 1 \) (repeats every 4).
- \( 15 \mod 4 = 3 \), so the unit digit of \( 3^{15} \) is \( 7 \).
- Thus:
\[ a_0 + a_1 + a_{30} = 1 + 30 + 7 = 38 \]

5. **Find the unit digit**:
- The unit digit of \( 38 \) is \( 8 \).

### Final Answer:
The digit at the unit place of \( a_0 + a_1 + a_{30} \) is \( 8 \).