Can someone check my answers and help me
1. A vector in standard position has its initial points at (0,0)
True?
2. The general formula for finding the probability of two events A and B that are not mutually exclusive and overlap is given by: P(A or B) = P(A) + P(B) + P(A and B)
True?
3. The formula for the arrangement of r of n objects is given by: n!/(n-r)!
False?
4. In buying a car, a customer has a choice of 3 bumper types, 4 stereo systems, 8 interior colors, 2 engines, 4 carpets, and 6 seat covers. How many different combinations of these 6 options are available?
Is it 4608?
5. The following statement is true by mathematical induction: n! > or equal to 2^n for all natural numbers n
False
Thank you!
1. ok
2. dependent or independent?
independent is P(A) + P(B) - P(A&B)
3. True - check your text
4. ok
5. ok
but it is true for n>3
So if it is true for n>3, would the answer be true?
NO! All natural numbers means n>0
1. Yes, that is true. A vector in standard position means that its initial point is located at the origin (0,0) on a coordinate plane.
2. No, that statement is not true. The correct formula for finding the probability of two events A and B that are not mutually exclusive and overlap is given by: P(A or B) = P(A) + P(B) - P(A and B). This formula subtracts the probability of the events occurring together (A and B) to account for the overlap.
3. No, that statement is false. The correct formula for the arrangement of r of n objects, also known as a permutation, is given by: n!/(n-r)!. In this formula, n! represents the factorial of n, which means multiplying all positive integers from 1 to n.
4. To find the number of different combinations, you need to multiply the number of options for each choice together. In this case, the number of different combinations is calculated as follows:
3 (bumper types) x 4 (stereo systems) x 8 (interior colors) x 2 (engines) x 4 (carpets) x 6 (seat covers) = 4608. Therefore, your answer is correct.
5. No, that statement is false. The statement n! ≥ 2^n is not true for all natural numbers n. To prove this, you can find a counterexample where the statement does not hold. For example, when n = 1, we have 1! = 1 which is not greater than or equal to 2^1 = 2. Therefore, the statement is false.
You're welcome! Let me know if there's anything else I can help you with.