a women student is to answer b10 out of 13 questions.find the number of her choices where she must answer
a,the frist two question
b,exactly 3 out of the frist 5 question
c,the first or second question but not both;
at least3 of the frist 5 question
To answer these questions, we will use the concepts of combinations and permutations.
a) The number of choices where the student must answer the first two questions:
To solve this, we need to select 2 questions out of the 2 questions in the first position. Since both questions are selected, there is only one possible combination. Therefore, the number of choices where the student must answer the first two questions is 1.
b) The number of choices where the student must answer exactly 3 out of the first 5 questions:
To solve this, we need to select 3 questions out of the 5 questions in the first position. Since the order of answering the questions doesn't matter, we will use combinations. The formula for combinations is nCr = n! / (r! * (n-r)!), where n is the total number of objects and r is the number of objects to be chosen.
In this case, n = 5 (the total number of questions in the first position) and r = 3 (the number of questions the student must answer). Plugging these values into the formula, we get:
5C3 = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = (5 * 4 * 3 * 2 * 1) / (3 * 2 * 1 * 2 * 1) = 10.
Therefore, the number of choices where the student must answer exactly 3 out of the first 5 questions is 10.
c) The number of choices where the student must answer the first or second question but not both:
To solve this, we will subtract the number of choices where the student must answer both the first and second question from the total number of choices where the student must answer at least one of the first two questions.
The total number of choices where the student must answer at least one of the first two questions is 2 (since there are 2 questions). The number of choices where the student must answer both the first and second question is 1 (as found in part a).
Therefore, the number of choices where the student must answer the first or second question but not both is 2 - 1 = 1.
d) The number of choices where the student must answer at least 3 out of the first 5 questions:
To solve this, we can use the principle of inclusion-exclusion. We need to find the sum of the choices where the student must answer 3, 4, or 5 out of the first 5 questions.
Using the formula for combinations as explained in part b, we can calculate the choices for each case:
- For answering 3 questions out of 5: 5C3 = 10 (as found in part b).
- For answering 4 questions out of 5: 5C4 = 5 (using the formula).
- For answering all 5 questions: 5C5 = 1 (using the same formula).
Adding up these choices, we get 10 + 5 + 1 = 16.
Therefore, the number of choices where the student must answer at least 3 out of the first 5 questions is 16.