A women student is to answer 10 out of 13 questions. Find the number of her choices where
she must answer: (a) the first two questions; (b) the first or second question but not both; (c)
exactly 3 out of the first 5 questions; (d) at least 3 of the first 5 questions.
(a) The student must answer the first two questions. There is only 1 choice for this.
(b) The student can choose to answer the first question but not the second, or answer the second question but not the first. This gives 2 choices.
(c) The student must answer exactly 3 out of the first 5 questions. This can be done in $\binom{5}{3}=10$ ways.
(d) The student can choose to answer any 3, 4, or 5 questions out of the first 5. This can be done in $\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=10+5+1=16$ ways.
Therefore:
(a) 1 choice
(b) 2 choices
(c) 10 choices
(d) 16 choices
To find the number of choices for each case, we can use combinations.
(a) She must answer the first two questions. This leaves 8 questions for her to choose from. The number of ways she can choose 8 questions out of 8 is given by the combination formula:
C(8, 8) = 1
(b) She must answer the first or second question but not both. This means she can choose either the first question (leaving 9 questions to choose from) or the second question (leaving 9 questions to choose from), but not both. The number of ways she can choose 9 questions out of 9 is given by the combination formula:
C(9, 9) = 1
So, the total number of choices for case (b) is:
C(1, 1) + C(1, 1) = 1 + 1 = 2
(c) She must answer exactly 3 out of the first 5 questions. This means she can choose 3 questions out of the 5 available. The number of ways she can choose 3 questions out of 5 is given by the combination formula:
C(5, 3) = (5! / (3! * (5-3)!)) = 10
(d) She must answer at least 3 out of the first 5 questions. This means she can choose 3, 4, or all 5 questions. The number of ways she can choose 3 questions is given by the combination formula:
C(5, 3) = 10
The number of ways she can choose 4 questions is given by the combination formula:
C(5, 4) = 5
The number of ways she can choose all 5 questions is given by the combination formula:
C(5, 5) = 1
So, the total number of choices for case (d) is:
C(5, 3) + C(5, 4) + C(5, 5) = 10 + 5 + 1 = 16
So, the number of choices for each case is:
(a) 1 choice
(b) 2 choices
(c) 10 choices
(d) 16 choices
To find the number of choices for each scenario, we can use the concept of combinations.
(a) To answer the first two questions, there is only one way to do so. Therefore, the number of choices is 1.
(b) To answer the first or second question but not both, we need to consider two cases:
Case 1: Answer the first question and not the second question.
Case 2: Answer the second question and not the first question.
In each case, there is only one way to answer the questions, so the number of choices is 1 + 1 = 2.
(c) To answer exactly 3 out of the first 5 questions, we need to choose any combination of 3 questions from the first 5. This can be calculated using combinations, denoted as C(n, r), where n is the total number of questions and r is the number of questions to be chosen.
Therefore, the number of choices is C(5, 3) = 5! / (3!(5-3)!) = 10.
(d) To answer at least 3 of the first 5 questions, we need to consider three cases:
Case 1: Answer 3 questions out of the first 5.
Case 2: Answer 4 questions out of the first 5.
Case 3: Answer all 5 questions.
For each case, we can calculate the number of choices using combinations:
Case 1: C(5, 3) = 5! / (3!(5-3)!) = 10.
Case 2: C(5, 4) = 5! / (4!(5-4)!) = 5.
Case 3: C(5, 5) = 5! / (5!(5-5)!) = 1.
Therefore, the number of choices is 10 + 5 + 1 = 16.
In summary:
(a) The number of choices is 1.
(b) The number of choices is 2.
(c) The number of choices is 10.
(d) The number of choices is 16.