1) Suppose of the 4,000 freshmen at a college everyone is enrolled in a mathematics or an English class during a given quarter. If 2,000 are enrolled in a mathematics class, and 3,000 in an English class, how many are enrolled in both a mathematics class and an English class?
2) At a manufacturing plant, a product goes through assembly, testing, and packing. If a plant has three assembly stations, two testing stations, and two packing stations, in how many different ways can a product achieve its completion?
3) In how many ways can a 10-question multiple choice test with four possible answers for each question be answered?
1) 2,000 + 3,000 - 4,000 = ?
2) If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
3 * 2 * 2 = ?
3) 4^10 = ?
1) To find the number of students enrolled in both a mathematics class and an English class, we can use the principle of inclusion-exclusion.
First, we know that the total number of freshmen is 4,000. Out of these, 2,000 are enrolled in a mathematics class and 3,000 are enrolled in an English class.
To find the number of students enrolled in both, we need to subtract the number of students enrolled only in mathematics and the number of students enrolled only in English from the total number of students.
Let's calculate each of these steps:
Number of students enrolled only in mathematics = 2,000 - (number enrolled in both)
Number of students enrolled only in English = 3,000 - (number enrolled in both)
Now, we can substitute these values back into the equation:
Total number of students = number enrolled only in mathematics + number enrolled only in English + number enrolled in both
4,000 = 2,000 - (number enrolled in both) + 3,000 - (number enrolled in both) + number enrolled in both
Simplifying the equation, we find:
4,000 = 5,000 - (number enrolled in both)
Now, we can solve for the number of students enrolled in both:
Number enrolled in both = 5,000 - 4,000 = 1,000
Therefore, there are 1,000 students enrolled in both a mathematics class and an English class.
2) To find the number of different ways a product can achieve its completion, we need to multiply the number of choices for each step.
The product goes through three assembly stations, so there are 3 choices for assembly.
It then goes through two testing stations, so there are 2 choices for testing.
Finally, it goes through two packing stations, so there are 2 choices for packing.
To find the total number of ways, we multiply these choices together:
Total number of ways = 3 (assembly) * 2 (testing) * 2 (packing)
Total number of ways = 12
Therefore, there are 12 different ways a product can achieve its completion.
3) To find the number of ways a 10-question multiple-choice test can be answered, we need to calculate the number of choices for each question and multiply them together.
For each question, there are 4 possible answers. Therefore, there are 4 choices for each question.
To find the total number of ways, we multiply the number of choices for each question together:
Total number of ways = 4^10 (four choices per question, ten questions)
Total number of ways = 1,048,576
Therefore, there are 1,048,576 ways a 10-question multiple-choice test can be answered.