To solve the problem, we need to determine the number of distinct combinations of answers that can earn a candidate a score from the set {15, 16, 17, 18, 19, 20} in a multiple-choice examination consisting of 10 questions. Each question has 4 options, with only one correct answer. The scoring system awards 3 marks for a correct answer and -1 mark for a wrong answer.
### Step-by-Step Solution:
1. **Define Variables**:
Let \( X \) be the number of questions answered correctly. Since there are 10 questions in total, the number of questions answered incorrectly will be \( 10 - X \).
2. **Calculate Total Score**:
The total score \( S \) can be expressed in terms of \( X \):
\[
S = 3X + (-1)(10 - X) = 3X - (10 - X) = 3X - 10 + X = 4X - 10
\]
3. **Set Up the Equation**:
We need to find the values of \( X \) such that the score \( S \) is in the set {15, 16, 17, 18, 19, 20}. Therefore, we set up the equations:
\[
4X - 10 = S
\]
for \( S = 15, 16, 17, 18, 19, 20 \).
4. **Solve for Each Score**:
- For \( S = 15 \):
\[
4X - 10 = 15 \implies 4X = 25 \implies X = 6.25 \quad \text{(not an integer)}
\]
- For \( S = 16 \):
\[
4X - 10 = 16 \implies 4X = 26 \implies X = 6.5 \quad \text{(not an integer)}
\]
- For \( S = 17 \):
\[
4X - 10 = 17 \implies 4X = 27 \implies X = 6.75 \quad \text{(not an integer)}
\]
- For \( S = 18 \):
\[
4X - 10 = 18 \implies 4X = 28 \implies X = 7 \quad \text{(valid integer)}
\]
- For \( S = 19 \):
\[
4X - 10 = 19 \implies 4X = 29 \implies X = 7.25 \quad \text{(not an integer)}
\]
- For \( S = 20 \):
\[
4X - 10 = 20 \implies 4X = 30 \implies X = 7.5 \quad \text{(not an integer)}
\]
5. **Valid Score**:
The only valid score is when \( S = 18 \) and \( X = 7 \). This means the candidate answered 7 questions correctly and 3 questions incorrectly.
6. **Calculate Combinations**:
The number of ways to choose 7 correct answers from 10 questions is given by the combination formula:
\[
\binom{10}{7} = \binom{10}{3} = \frac{10!}{7!3!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]
### Final Answer:
The number of distinct combinations of answers that can earn the candidate a score from the set {15, 16, 17, 18, 19, 20} is \( \boxed{120} \).
