Key Concepts and Formulas

• Combinations: The number of ways to choose r items from a set of n items (where order doesn't matter) is given by the binomial coefficient: ${n \choose r} = \frac{n!}{r!(n-r)!}$
• Fundamental Principle of Counting: If there are m ways to do one thing and n ways to do another, then there are m × n ways to do both.

Step-by-Step Solution

Step 1: Select the four questions to be answered correctly.

• We need to choose 4 questions out of 6 to answer correctly. The order in which we choose them doesn't matter, so we use combinations.
• The number of ways to choose 4 questions out of 6 is: ${6 \choose 4} = \frac{6!}{4!(6-4)!} = \frac{6!}{4!2!} = \frac{6 \times 5}{2 \times 1} = 15$

Step 2: Determine the number of ways to answer the chosen four questions correctly.

• For each of the 4 chosen questions, there is only 1 correct answer out of the 4 options.
• Therefore, the number of ways to answer each of the 4 questions correctly is 1.
• The total number of ways to answer all 4 questions correctly is: $1 \times 1 \times 1 \times 1 = 1^4 = 1$

Step 3: Determine the number of ways to answer the remaining two questions incorrectly.

• For each of the remaining 2 questions, there are 3 incorrect answers out of the 4 options.
• Therefore, the number of ways to answer each of the 2 questions incorrectly is 3.
• The total number of ways to answer both questions incorrectly is: $3 \times 3 = 3^2 = 9$

Step 4: Calculate the total number of ways to answer exactly four questions correctly.

• Using the fundamental principle of counting, we multiply the number of ways to choose the 4 correct questions, the number of ways to answer those 4 questions correctly, and the number of ways to answer the remaining 2 questions incorrectly.
• Therefore, the total number of ways is: ${6 \choose 4} \times 1^4 \times 3^2 = 15 \times 1 \times 9 = 135$

Common Mistakes & Tips

• Remember to use combinations ( ${n \choose r}$ ) when the order of selection doesn't matter. Permutations (nPr) are used when order matters.
• Be careful to distinguish between the number of correct answers and the number of incorrect answers for each question.
• Always check your calculations and reasoning to avoid simple errors.

Summary

To find the number of ways a candidate can answer exactly four out of six multiple-choice questions correctly, we first determine the number of ways to choose the four questions to be answered correctly using combinations. Then, we determine the number of ways to answer those four questions correctly (which is 1 for each question). Finally, we determine the number of ways to answer the remaining two questions incorrectly (3 for each question). Multiplying these values together gives the total number of ways, which is 135.

Final Answer

The final answer is \boxed{135}.