A student is to answer 10 out of 13 questions in an exam. how many choices he has if he must answer at least 3 of the first
To find the number of choices the student has, we need to consider two scenarios:
Scenario 1: The student answers exactly 3 questions from the first 4 questions.
Scenario 2: The student answers more than 3 questions from the first 4 questions.
Scenario 1:
For this scenario, the student needs to choose 3 out of the first 4 questions. This can be calculated using the combination formula (nCr):
Number of ways to choose 3 questions from 4 = 4C3 = 4
Scenario 2:
For this scenario, the student needs to choose more than 3 questions from the first 4 questions. Since the student must answer at least 3 of the first 4 questions, the possible combinations are:
- Choose 4 questions: 1 way
- Choose 5 questions: 4C1 = 4 ways
- Choose 6 questions: 4C2 = 6 ways
Now, we can calculate the total number of choices for Scenario 2:
Total choices for Scenario 2 = 1 + 4 + 6 = 11
To find the total number of choices for the student, we can simply add the choices from both scenarios:
Total choices = Choices for Scenario 1 + Choices for Scenario 2
= 4 + 11
= 15
Therefore, the student has 15 choices to answer 10 out of 13 questions in the exam, provided that they must answer at least 3 of the first 4 questions.
To find the number of choices the student has, we need to consider the two cases separately:
Case 1: The student answers exactly 3 questions from the first 3 questions.
In this case, the student must answer exactly 3 out of the first 3 questions and choose the remaining 7 questions from the remaining 10 questions. Therefore, the number of choices in this case is given by:
C(3,3) * C(10,7) = 1 * 120 = 120
Case 2: The student answers more than 3 questions from the first 3 questions.
In this case, let's assume the student answers k questions (where k is greater than 3) from the first 3 questions. The remaining (10 - k) questions can then be chosen from the remaining 10 questions. Summing up the choices for each possible value of k will give us the total number of choices:
Σ (k=4 to 10) [ C(3,k) * C(10,10-k) ]
To calculate this sum, we can use the formula for the binomial coefficient: C(n,r) = n!/[(n-r)! * r!]
Plugging in the values and calculating the sum using a calculator or a spreadsheet will give us the final answer.
So, the total number of choices the student has is the sum of Case 1 and Case 2.