# Combinatorics

 1. (28 p.) There are 27 candidates in elections and $$n$$ citizens that vote for them. If a candidate gets $$m$$ votes, then $$100m/n \leq m-1$$. What is the smallest possible value of $$n$$?

 2. (26 p.) In a tournament club $$C$$ plays 6 matches, and for each match the probabilities of a win, draw and loss are equal. If the probability that $$C$$ finishes with more wins than losses is $$\frac pq$$ with $$p$$ and $$q$$ coprime $$(q>0)$$, find $$p+q$$.

 3. (4 p.) Let $$S$$ be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to $$S$$.

 4. (11 p.) Assume that $$A$$ is a 40-element subset of $$\{1,2,3,\dots,50\}$$, and let $$n$$ be the sum of the elements of $$A$$. Find the number of possible values of $$n$$.

 5. (28 p.) We are given an unfair coin. When the coin is tossed, the probability of heads is 0.4. The coin is tossed 10 times. Let $$a_n$$ be the number of heads in the first $$n$$ tosses. Let $$P$$ be the probability that $$a_n/n \leq 0.4$$ for $$n = 1, 2, \dots , 9$$ and $$a_{10}/10 = 0.4$$. Evaluate $$\frac{P\cdot 10^{10}}{24^4}$$.

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