Key Concepts and Formulas

• Combinations: The number of ways to choose $r$ items from a set of $n$ distinct items, where order doesn't matter, is given by ${}^n C_r = \frac{n!}{r!(n-r)!}$.
• Sum Rule (Addition Principle): If there are $m$ ways to do one thing and $n$ ways to do another, and these things are mutually exclusive, then there are $m + n$ ways to do either.
• Product Rule (Multiplication Principle): If there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both.

Step-by-Step Solution

Step 1: Define the groups of questions.

We divide the 13 questions into two groups based on the problem's constraint:

• Group 1: The first 5 questions.
• Group 2: The remaining 8 questions.

This division helps us organize the selection process according to the given condition.

Step 2: Identify the possible scenarios.

The student must choose at least 4 questions from Group 1. This means they can choose exactly 4 or exactly 5 questions from Group 1. These scenarios are mutually exclusive.

• Scenario 1: Choosing exactly 4 questions from Group 1.
• Scenario 2: Choosing exactly 5 questions from Group 1.

Step 3: Analyze Scenario 1: Choosing exactly 4 questions from Group 1.

• Step 3.1: Choose 4 questions from Group 1. We need to select 4 questions out of the first 5. The number of ways to do this is given by ${}^5 C_4$. ${}^5 C_4 = \frac{5!}{4!(5-4)!} = \frac{5!}{4!1!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(4 \times 3 \times 2 \times 1)(1)} = 5$ There are 5 ways to choose 4 questions from the first 5.

• Step 3.2: Choose the remaining questions from Group 2. Since the student must answer 10 questions in total, and 4 have been chosen from Group 1, we need to choose $10 - 4 = 6$ questions from Group 2 (the remaining 8 questions). The number of ways to do this is given by ${}^8 C_6$. ${}^8 C_6 = \frac{8!}{6!(8-6)!} = \frac{8!}{6!2!} = \frac{8 \times 7 \times 6!}{6! \times 2 \times 1} = \frac{8 \times 7}{2} = 28$ There are 28 ways to choose 6 questions from the remaining 8.

• Step 3.3: Calculate the total number of ways for Scenario 1. Using the Product Rule, we multiply the number of ways to choose from Group 1 and Group 2. $\text{Ways for Scenario 1} = {}^5 C_4 \times {}^8 C_6 = 5 \times 28 = 140$

Step 4: Analyze Scenario 2: Choosing exactly 5 questions from Group 1.

• Step 4.1: Choose 5 questions from Group 1. We need to select 5 questions out of the first 5. The number of ways to do this is given by ${}^5 C_5$. ${}^5 C_5 = \frac{5!}{5!(5-5)!} = \frac{5!}{5!0!} = 1$ There is only 1 way to choose all 5 questions from the first 5.

• Step 4.2: Choose the remaining questions from Group 2. Since the student must answer 10 questions in total, and 5 have been chosen from Group 1, we need to choose $10 - 5 = 5$ questions from Group 2 (the remaining 8 questions). The number of ways to do this is given by ${}^8 C_5$. ${}^8 C_5 = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = \frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} = \frac{8 \times 7 \times 6}{6} = 56$ There are 56 ways to choose 5 questions from the remaining 8.

• Step 4.3: Calculate the total number of ways for Scenario 2. Using the Product Rule, we multiply the number of ways to choose from Group 1 and Group 2. $\text{Ways for Scenario 2} = {}^5 C_5 \times {}^8 C_5 = 1 \times 56 = 56$

Step 5: Calculate the total number of choices.

Since Scenario 1 and Scenario 2 are mutually exclusive, we add the number of ways from each scenario using the Sum Rule. $\text{Total number of choices} = \text{Ways for Scenario 1} + \text{Ways for Scenario 2} = 140 + 56 = 196$

Common Mistakes & Tips

• Understanding "at least": "At least 4" means 4 or more. Be sure to consider all valid cases.
• Applying combination formula correctly: Ensure you are using the correct values for $n$ and $r$ in the combination formula. Double-check your calculations.
• Remembering $0! = 1$: This is a common source of error when calculating combinations.

Summary

We divided the problem into two mutually exclusive scenarios based on the constraint of choosing at least 4 questions from the first 5. We calculated the number of ways for each scenario using combinations and the product rule. Finally, we added the results to find the total number of choices.

The final answer is $\boxed{196}$, which corresponds to option (C).