To solve this problem, we need to calculate the number of ways the student can select 8 questions while satisfying the given constraints.

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Constraints:

1. She must answer at least 3 of the last 5 questions.
2. She can answer at most 4 of the first 5 questions.

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Step-by-step Solution:

1. Total choices:

   • First 5 questions → Let $x$ be the number of questions answered from the first 5.
   • Last 5 questions → She must answer $8 - x$ questions from the last 5.

2. Constraint on the last 5 questions:

   • At least 3 questions must be answered from the last 5, so $8 - x \geq 3$, which implies $x \leq 5$.

3. Constraint on the first 5 questions:

   • At most 4 questions can be answered from the first 5, so $x \leq 4$.

Combining these two constraints, $x \leq 4$.

4. Possible values of $x$:
   • Since $8 - x \leq 5$ (the maximum number of questions available in the last 5), $x \geq 3$.
   • Thus, $x$ can be $3$ or $4$.

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Case Analysis:

For each $x$, calculate the number of ways to choose the questions.

Case 1: $x = 3$ (3 from the first 5, 5 from the last 5)

• Number of ways to choose 3 questions from the first 5: $\binom{5}{3} = 10$
• Number of ways to choose 5 questions from the last 5: $\binom{5}{5} = 1$
• Total for this case: $10 \times 1 = 10$

Case 2: $x = 4$ (4 from the first 5, 4 from the last 5)

• Number of ways to choose 4 questions from the first 5: $\binom{5}{4} = 5$
• Number of ways to choose 4 questions from the last 5: $\binom{5}{4} = 5$
• Total for this case: $5 \times 5 = 25$

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Final Answer:

Add the totals from all cases: $10 + 25 = 35$

The student can choose the questions in 35 ways.

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Do you want a detailed breakdown of any specific step or additional explanation?

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Related Questions:

1. What happens if the student must answer at least 4 of the last 5 questions instead of 3?
2. How does the number of combinations change if the student is allowed to answer all 10 questions?
3. What is the total number of ways to select 8 questions without any restrictions?
4. How would the result change if the student could answer at most 3 of the first 5 questions?
5. What if the student must answer exactly 2 questions from the last 5?

Tip:

When dealing with multiple constraints in combinatorics, break the problem into smaller cases and systematically calculate for each case.