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Math/Stat 304 Special Quiz 17 February 2014 Name: ______________________ Instructions: Do not simplify your answers --- particularly when they involve factorials, combinations and/or permutations. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities. - George Boole, An Investigation of the Laws of Thought 2 Note: Do not simplify your answers --- particularly when they involve factorials, combinations and/or permutations. The word “subset”, unless otherwise qualified, means “unordered subset.” The word “deck” refers to a standard deck of cards. 1. How many rearrangements are there of the word PROBABILISTIC? 2. What is the total number of subsets of a set of 3000 distinguishable llamas? (Express your answer in closed form.) 3. What is the total number of subsets of a set of 100 indistinguishable tootsie rolls? 4. How many subsets of 8 cards from a deck of 52 contain exactly 4 distinct pairs? (Note “4 of a kind” does not count as two pairs.)? 5. How many subsets of 5 cards from a deck of 52 contain no pair? 3 6. How many subsets of 6 cards from a deck of 52 contain exactly 3-of-a-kind but not an additional pair (nor 4-of-a-kind or a second 3-of-a-kind)? 7. How many subsets of 10 cards from a deck of 52 contain exactly 5 red cards and exactly 5 black cards? 8. Inmates at the Oz Penitentiary wear either Blue uniforms or Red uniforms. Thirty-five prisoners line up to go to sports activities; 15 of them wear Red uniforms and the rest wear Blue uniforms. How many possible lines can be formed if no two inmates wearing Red uniforms are allowed to stand side-by-side? 9. A red die, a blue die, and a yellow die (all 6 sided) are rolled. How many outcomes are there if the number on the red die is (strictly) less than the number on the blue die, which in turn is (strictly) less than the number on the yellow die? 4 10. In how many ways can 5 people {A, B, C, D, E) line up if there must be exactly one person between A and B? 11. An urn contains 5 distinguishable Red balls, 7 distinguishable Blue balls and 8 distinguishable White balls. In how many ways can one select an (unordered) subset of 8 balls containing 3 Red, 3 Blue and 2 White balls? 12. A four-sided die is flipped 20 times. (Here, order counts.) In how many ways can one obtain exactly 8 ones, 5 twos, 4 threes and 3 fours? 13. In how many ways can you be dealt 6 cards (where order matters) in such a way that the second ace appears on the 6th (and last) card? 5 14. An elevator starts at the first floor with 17 people (not including Albertine, the elevator operator) and discharges them all by the time it reaches the eleventh floor. In how many different ways could Albertine have perceived the people leaving the elevator if all people look alike to her? (Note: nobody exits on the first floor.) 15. A drawer contains 11 pairs of gloves (each pair of a different style and color). In how many ways can one choose (an unordered set of) six gloves that includes exactly two pairs of gloves of the same style and color? 16. Vladimir, at point A, must meet Estragon at point B. But first Vladimir must stop at the hardware store, point C, to purchase some rope. In how many ways can Vladimir achieve this journey if he takes the shortest route in each of the two sub-trips? 6 17. The Astrology Department of the University of Oz offers nine (different) 100-level courses, ten (different) 200-level courses, and thirteen (different) 300-level courses. A fulltime student must select two 100-level courses and three upper-level (meaning 200-level or 300-level) courses with the stipulation that she must have at least one 200-level course and at least one 300-level course. If Odette wishes to be a full-time student, in how many ways can she choose her curriculum?