On a certain examination the student must answer 8 out of 12 questions including exactly 5 of the first six. In how many ways may he write the examination.?
well, there are 6 ways to answer 5 of the 1st 6 correctly (the same as the number of ways to get one of them wrong)
That leaves 6 questions, of which 4 can be chosen in 6C4 = 15 ways.
Each set of 4 can be answered in 2^4 ways.
Well, let me do some mental calculations while juggling rubber chickens and unicycling at the same time. Ahem, let's see...
To solve this problem, we need to consider two scenarios: choosing the first six questions and choosing the remaining two questions.
For the first scenario, the student needs to choose exactly 5 out of the first six questions. This can be calculated using the combination formula, which is like trying to figure out how many ways you can arrange those six questions in groups of 5. So, it's C(6,5) = 6.
For the second scenario, the student needs to choose 2 out of the remaining 6 questions. Again, we can use the combination formula to find the number of ways, which is C(6,2) = 15.
To find the total number of ways the student can write the examination, we multiply the number of ways for each scenario together: 6 x 15 = 90.
So, there you have it! The student can write the examination in 90 different ways. Now, if only I could find a rubber chicken mathematician to double-check my calculations... 😄
To find the number of ways the student can write the examination, we need to consider the two requirements: answering 8 out of 12 questions and 5 out of the first 6.
Step 1: Choose 5 out of the first 6 questions
The student must answer exactly 5 out of the first 6 questions. This can be done in the following way: C(6, 5) = 6 (choosing 5 questions out of 6).
Step 2: Choose 3 out of the remaining 6 questions
Since the student needs to answer a total of 8 questions, and has already chosen 5 out of the first 6, they need to choose 3 more questions from the remaining 6. This can be done in the following way: C(6, 3) = 20 (choosing 3 questions out of 6).
Step 3: Multiply the possibilities from Step 1 and Step 2
To find the total number of ways the student can write the examination, we multiply the possibilities from Step 1 and Step 2: 6 x 20 = 120.
Therefore, there are 120 ways the student can write the examination.
To find the number of ways the student can write the examination, we can break down the problem into separate cases and then add up the possibilities.
Case 1: Choosing exactly 5 questions from the first six
There are 6 questions in the first six. The student needs to choose exactly 5 of them. This can be done in "6 choose 5" ways, which is denoted by C(6, 5) or 6C5. The formula for this is:
C(n, r) = n! / (r! * (n - r)!)
Using this formula, we can calculate the number of ways to choose 5 questions from 6:
C(6, 5) = 6! / (5! * (6 - 5)!) = 6
Case 2: Choosing 3 questions from the remaining 6
After selecting 5 questions from the first six, we have 1 remaining question to choose from the first six and 6 questions left from the remaining six. The student needs to select 3 questions out of these 6. This can be done in "6 choose 3" ways:
C(6, 3) = 6! / (3! * (6 - 3)!) = 20
Case 3: Choosing 8 questions from the overall set of 12
At this point, the student has chosen 5 questions from the first six and 3 questions from the remaining six. Now, the student has to choose 8 questions from the remaining set of 12 (excluding the 6 questions already chosen).
C(12, 8) = 12! / (8! * (12 - 8)!) = 495
Finally, we multiply the number of possibilities in each case to get the total number of ways the student can write the examination:
Total number of ways = C(6, 5) * C(6, 3) * C(12, 8) = 6 * 20 * 495 = 59,400
Therefore, the student can write the examination in 59,400 ways.