1. Let the numbers of correct, wrong, and unattempted questions be $c,w,u$ respectively.

   Since there are 5 questions, $c+w+u=5.$

2. Write the score equation.

   Each correct answer gives $3$ marks, each wrong answer gives $-2$ marks, and each unattempted question gives $0$ marks.

   So total score is $3c-2w=5.$

3. Find non-negative integer solutions of $3c-2w=5.$

   Rearranging, $3c=5+2w.$

   We test non-negative integers with $c+w\le 5$:

   • If $w=0$, then $3c=5$ impossible.
   • If $w=1$, then $3c=7$ impossible.
   • If $w=2$, then $3c=9 \Rightarrow c=3$.
   • If $w=3$, then $3c=11$ impossible.
   • If $w=4$, then $3c=13$ impossible.
   • If $w=5$, then $3c=15 \Rightarrow c=5$, but then score is $15-10=5$ and $c+w=10$, impossible since only 5 questions.

   So the only valid solution is $c=3,\quad w=2,\quad u=0.$

4. Count the number of ways.

   First choose which 3 of the 5 questions are answered correctly: $\binom{5}{3}=10.$

   The remaining 2 questions must be answered wrongly.

   For each wrongly answered question, there are 2 wrong choices available (since each question has 3 choices and exactly 1 is correct).

   Hence number of ways for the 2 wrong answers is $2^2=4.$

5. Total number of ways $\binom{5}{3}\cdot 2^2 = 10\cdot 4 = 40.$

Therefore, the number of ways is $\boxed{40}.$