1. Let the number of questions selected from sections $A, B, C$ be $a,b,c$ respectively.

   Then $a+b+c=15$ with constraints:

   • $4 \le a \le 8$
   • $4 \le b \le 6$
   • $4 \le c \le 6$

2. Since each section must contribute at least $4$ questions, put $a=4+x,\quad b=4+y,\quad c=4+z$ where $x\ge 0,\ y\ge 0,\ z\ge 0.$

   Then $x+y+z=15-12=3.$

3. Also, because section sizes are limited:

   • $a\le 8 \Rightarrow x\le 4$
   • $b\le 6 \Rightarrow y\le 2$
   • $c\le 6 \Rightarrow z\le 2$

4. Now find all nonnegative integer solutions of $x+y+z=3$ satisfying $y\le 2$ and $z\le 2$.

   The possible $(x,y,z)$ are: $(3,0,0), (2,1,0), (2,0,1), (1,2,0), (1,1,1), (1,0,2), (0,2,1), (0,1,2).$

   Hence corresponding $(a,b,c)$ are:

   $$(7,4,4), (6,5,4), (6,4,5), (5,6,4), (5,5,5), (5,4,6), (4,6,5), (4,5,6).$$

5. For each distribution, count the number of ways:

   • For $(7,4,4)$: $\binom{8}{7}\binom{6}{4}\binom{6}{4}=8\cdot 15\cdot 15=1800$

   • For $(6,5,4)$: $\binom{8}{6}\binom{6}{5}\binom{6}{4}=28\cdot 6\cdot 15=2520$

   • For $(6,4,5)$: $\binom{8}{6}\binom{6}{4}\binom{6}{5}=28\cdot 15\cdot 6=2520$

   • For $(5,6,4)$: $\binom{8}{5}\binom{6}{6}\binom{6}{4}=56\cdot 1\cdot 15=840$

   • For $(5,5,5)$: $\binom{8}{5}\binom{6}{5}\binom{6}{5}=56\cdot 6\cdot 6=2016$

   • For $(5,4,6)$: $\binom{8}{5}\binom{6}{4}\binom{6}{6}=56\cdot 15\cdot 1=840$

   • For $(4,6,5)$: $\binom{8}{4}\binom{6}{6}\binom{6}{5}=70\cdot 1\cdot 6=420$

   • For $(4,5,6)$: $\binom{8}{4}\binom{6}{5}\binom{6}{6}=70\cdot 6\cdot 1=420$

6. Add all these values: $1800+2520+2520+840+2016+840+420+420=11376.$

Therefore, the total number of ways is $\boxed{11376}.$