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Assn1 - round table so that none of the races forms a...

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1. Consider a set of objects, each of which may satisfy one or more of four properties p 1 , p 2 , p 3 and p 4 . Write a formula for the number of objects that satisfy exactly one or exactly three of these properties (using only the terms of form N ( p i 1 p i 2 . . . ) that appear in the principle of inclusion and exclusion). 2. 100 persons hold an election, voting for one of 5 candidates. Each person gives his/her vote to one of the candidates. How many results can the election have, if we know that each of the candidates received at least 10, but at most 30 votes? The election is anonymous, so only the number of votes each candidate receives is relevant, not who exactly voted for him. 3. In how many ways can 4 dwarves, 3 elves and 5 orcs sit around a
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Unformatted text preview: round table, so that none of the races forms a contiguous block? The creatures are distinguishable (i.e., the order of say individual dwarves is important). Also, the seating arrangements differing only by a rotation of the table are considered to be distinct. 4. Write a general formula for the number of integers between 1 and n (inclusive) that are divisible by at least one of a 1 , a 2 , . . . , a k , where a 1 , . . . , a k are primes. 5. You permute the integers 1, . . . , 10 randomly. What is the probability that no integer i will appear on the i-th position, and furthermore, 1 will not appear on the 5-th position?...
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