- - 1. Consider lists made from the letters C, O, M, P, U, T, E, R. (a) How many lists of length 5 are there if repetition is allowed? (b) How many lists of length 5 are there if repetition is not allowed? (c) How many lists of length 4 are there that begin with the letter P if repetition is allowed? (d) How many lists of length 4 are there that do not begin with the letter P if repetition is not allowed? (e) How many lists of length 3 are there if repetition is allowed and the list must contain the letter C? - (f) How many lists of length 3 are there if repetition is not allowed and the list must contain the letter C? (Use MP/AP/SP) [4P] 2. How many integers between 1 and 1000 are divisible by 5? How many are not? (Use common sense) [1P] 3. A recent survey of European households revealed that 97% have at least one DVD-player, 96% have a cell phone, and 93% have a cell phone and at least one DVD-player. What percentage of households in the European Union have neither a cell phone nor a DVD-player? (Use IEP) [1P] 4. How many ways are there to select a first-prize winner, a second-prize winner, and a third- prize winner from among 250 students who have entered a campus-wide contest? (Use P/C) [1P] 5. How many different committees of three students can be formed from a group of five students? (Use P/C) [1P] 6. True or False? According to the Pigeonhole Principle, if we have n marbles and k boxes, then each box can have at most (7) marbles. [1P] 7. Consider a random group of 1,000 people. At least how many people were born in the same month? (Use DP/PP) [1P] 8. Students can score on the final exam anywhere from zero to 100 points. How many students need to take the exam so that it is guaranteed that at least two students have the exact same score? (Use DP/PP; watch out: how many different scores are there?) [1P] 9. Suppose students can earn one out of the five letter grades A, B, C, D, and F. How many students must a class have so that at least six of them will receive the same grade? (Use DP/PP) [1P]