i) If you have twenty ones and fifteen zeroes how many ways can we arrange these digits such that no two zeroes are adjacent? ii) Show that the number of subsets of a finite set S is 2 |S| i) [The Birthday Problem] What is the probability that at least one of ten people are born on the same day of the year? [Note: as the number of people increase a surprising result occurs] ** [Hint: What is the probability ALL have different birthdays?] iv) You have Eight Boys and Seven Girls. How many ways can you arrange three boys and two girls for a picture? ** (Hint: consider first the arrangement positions 1, 2,3 are boys and 4,5 are girls ** how many ways cant this be done Next: How many ways can we CHOOSE three positions out of 1,2,3,4,5 to put boys and the rest to put girls?]

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