Assignment:
Q1. How many ways are there to choose 5 items (with repetition) from among 7 varieties? Explain, as precisely as possible, why we should not factor out by the number of arrangements of the 5 choices. ( Don't prove that the answer is wrong-we already did that- explain why it's wrong, i.e., why we shouldn't factor out by r!) It might be helpful to think of a smaller case: 2 choices from among 3 varieties. Following this method, we could count 3 2/2! ways. But if you write down all 9 permutations.
Q2. How many positive integers less than 1,000 are a) divisible by 7, b) are divisible by both 7 and 11 c) have distinct digits
Q3. How many bit strings of length 10 contain either five consecutive 0s or five consecutive 1s.
Q4. Use the product rule to show that there are 2^(2^n) different truth tables for propositions of n variables
Q5. How many permutations of the letters ABCDEFGH contain:
a) the string ED
b) the strings BCA and ABF
Q6. How many solutions are there to the inequality x1 + x2 + x3 <= 11 where x1, x2, and x3 are non-negative integers?
Q7. How many different strings can be made from the letters in MISSISSIPPI using all the letters?
Q8. A shelf holds 12 books in a row. how many ways are there to choose five books so that no two adjacent books are chosen?
Q9. Show that if E and F are independent events, than Ebar and Fbar are also independent events.
Q10. Find the smallest number of people in a room so that the probability that someone has a birthday today exceeds 1/2 (assume all birthdays are equally likely and that the year has 366 days).
Q11. Find each of hte following probabilities when n independent Bernoulli trials are carried out with probability of success p:
a) the probability of no successes
b) the probability of at least one success
c) the probability of at most one success
d) the probability of at least two successes
Provide complete and step by step solution for the question and show calculations and use formulas.