Find the coefficient of `x^7` in the expansion of `(1+2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7x^6 + 8x^7)^10`

Answer

To find the coefficient of \( x^7 \) in the expansion of \( (1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7x^6 + 8x^7)^{10} \), we can use the multinomial expansion. We will consider different combinations of terms that can yield \( x^7 \).

### Step-by-Step Solution:

1. **Identify the terms contributing to \( x^7 \)**:
- We can have:
- 7 from \( 2x \) and 3 from \( 1 \).
- 1 from \( 8x^7 \) and 9 from \( 1 \).

2. **Calculate the contribution from the first case (7 from \( 2x \) and 3 from \( 1 \))**:
- We choose \( 7 \) terms of \( 2x \) and \( 3 \) terms of \( 1 \).
- The number of ways to choose \( 7 \) terms from \( 10 \) is given by \( \binom{10}{7} \).
- The contribution to the coefficient from \( (2x)^7 \) is \( 2^7 \).
- Therefore, the contribution from this case is:
\[ \binom{10}{7} \cdot 2^7 \cdot 1^3 = \binom{10}{7} \cdot 2^7 \]

3. **Calculate the contribution from the second case (1 from \( 8x^7 \) and 9 from \( 1 \))**:
- We choose \( 1 \) term of \( 8x^7 \) and \( 9 \) terms of \( 1 \).
- The number of ways to choose \( 1 \) term from \( 10 \) is given by \( \binom{10}{1} \).
- The contribution to the coefficient from \( (8x^7)^1 \) is \( 8^1 \).
- Therefore, the contribution from this case is:
\[ \binom{10}{1} \cdot 8^1 \cdot 1^9 = \binom{10}{1} \cdot 8 \]

4. **Combine the contributions**:
- The total coefficient of \( x^7 \) is the sum of the contributions from both cases:
\[ \text{Total Coefficient} = \binom{10}{7} \cdot 2^7 + \binom{10}{1} \cdot 8 \]

5. **Calculate the binomial coefficients and powers**:
- \( \binom{10}{7} = \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \)
- \( 2^7 = 128 \)
- \( \binom{10}{1} = 10 \)
- So, we have:
\[ \text{Total Coefficient} = 120 \cdot 128 + 10 \cdot 8 \]

6. **Perform the calculations**:
- \( 120 \cdot 128 = 15360 \)
- \( 10 \cdot 8 = 80 \)
- Therefore:
\[ \text{Total Coefficient} = 15360 + 80 = 15440 \]

### Final Answer:
The coefficient of \( x^7 \) in the expansion is \( \boxed{15440} \).