Question 1210232
Solution:
Let a partition of 17 with at least three parts be $\lambda_1 \ge \lambda_2 \ge \lambda_3 \ge \dots \ge \lambda_k > 0$, where $k \ge 3$ and $\sum_{i=1}^k \lambda_i = 17$.
The conditions on the largest, second-largest, third-largest, and fourth-largest parts are:
$\lambda_1 \ge 4$
$\lambda_2 \ge 3$
$\lambda_3 \ge 2$
$\lambda_4 \ge 1$ (if $k \ge 4$)

Consider a partition of 17 into $k$ parts satisfying these conditions.
Let $\lambda_1 = a_1 + 4, \lambda_2 = a_2 + 3, \lambda_3 = a_3 + 2, \lambda_i = a_i + 1$ for $i \ge 4$, where $a_1 \ge a_2 \ge a_3 \ge a_4 \ge \dots \ge a_k \ge 0$.

Case 1: Exactly 3 parts ($k=3$)
$\lambda_1 \ge \lambda_2 \ge \lambda_3 > 0$
$\lambda_1 \ge 4, \lambda_2 \ge 3, \lambda_3 \ge 2$
Let $\lambda_1 = x+4, \lambda_2 = y+3, \lambda_3 = z+2$, where $x \ge y \ge z \ge 0$.
$(x+4) + (y+3) + (z+2) = 17 \implies x+y+z = 8$.
Partitions of 8 into 3 non-negative integers:
(8, 0, 0), (7, 1, 0), (6, 2, 0), (6, 1, 1), (5, 3, 0), (5, 2, 1), (4, 4, 0), (4, 3, 1), (4, 2, 2), (3, 3, 2).
These correspond to partitions of 17: 12+3+2, 11+4+2, 10+5+2, 10+4+3, 9+6+2, 9+5+3, 8+7+2, 8+6+3, 8+5+4, 7+6+4. (10 partitions)

Case 2: Exactly 4 parts ($k=4$)
$\lambda_1 \ge 4, \lambda_2 \ge 3, \lambda_3 \ge 2, \lambda_4 \ge 1$.
Let $\lambda_1 = x+4, \lambda_2 = y+3, \lambda_3 = z+2, \lambda_4 = w+1$, where $x \ge y \ge z \ge w \ge 0$.
$(x+4) + (y+3) + (z+2) + (w+1) = 17 \implies x+y+z+w = 7$.
Partitions of 7 into 4 non-negative integers:
(7, 0, 0, 0), (6, 1, 0, 0), (5, 2, 0, 0), (5, 1, 1, 0), (4, 3, 0, 0), (4, 2, 1, 0), (4, 1, 1, 1), (3, 3, 1, 0), (3, 2, 2, 0), (3, 2, 1, 1), (2, 2, 2, 1). (11 partitions)

Case 3: Exactly 5 parts ($k=5$)
$\lambda_1 \ge 4, \lambda_2 \ge 3, \lambda_3 \ge 2, \lambda_4 \ge 1, \lambda_5 \ge 1$.
$x+4+y+3+z+2+w+1+v+1 = 17 \implies x+y+z+w+v = 6$. (7 partitions)

Case 4: Exactly 6 parts ($k=6$)
Sum of min parts = $4+3+2+1+1+1 = 12$. $x+y+z+w+v+u = 5$. (5 partitions)

Case 5: Exactly 7 parts ($k=7$)
Sum of min parts = $4+3+2+1+1+1+1 = 13$. $x+y+z+w+v+u+t = 4$. (5 partitions)

Case 6: Exactly 8 parts ($k=8$)
Sum of min parts = $4+3+2+1+1+1+1+1 = 14$. $x+y+z+w+v+u+t+s = 3$. (3 partitions)

Case 7: Exactly 9 parts ($k=9$)
Sum of min parts = $4+3+2+1+1+1+1+1+1 = 15$. $x+...+r = 2$. (2 partitions)

Case 8: Exactly 10 parts ($k=10$)
Sum of min parts = $4+3+2+1+1+1+1+1+1+1 = 16$. $x+...+q = 1$. (1 partition)

Case 9: Exactly 11 parts ($k=11$)
Sum of min parts = $4+3+2+1+1+1+1+1+1+1+1 = 17$. $x+... = 0$. (1 partition)

Total = $10 + 11 + 7 + 5 + 5 + 3 + 2 + 1 + 1 = 45$.

Final Answer: The final answer is $\boxed{45}$