© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Understand the Problem
The student has 13 questions in total and must answer exactly 10 of them, with the condition that at least 4 of these 10 answers must come from the first 5 questions.
Step 2: Break Down the Constraint
Since at least 4 questions must be chosen from the first 5, there are two possible scenarios:
Choose 4 questions out of the first 5.
Choose all 5 questions out of the first 5.
Step 3: Case-by-Case Combination Counts
Case 1: Select 4 from the first 5, and the remaining 6 questions (to make a total of 10) from the remaining 8 questions.
The number of ways to do this is:
$ C(5, 4) \times C(8, 6) $.
Case 2: Select all 5 questions from the first 5, and the remaining 5 questions (to make a total of 10) from the remaining 8 questions.
The number of ways to do this is:
$ C(5, 5) \times C(8, 5) $.
Step 4: Calculate Each Case
$ C(5, 4) = 5 $,
$ C(8, 6) = 28 $, so Case 1 total = $ 5 \times 28 = 140 $.
$ C(5, 5) = 1 $,
$ C(8, 5) = 56 $, so Case 2 total = $ 1 \times 56 = 56 $.
Step 5: Sum Up All Possible Ways
Total number of ways = $ 140 + 56 = 196 $.
Step 6: Final Answer
Therefore, the number of choices available to the student is 196.
