# Combinatorics

 1. (6 p.) Let $$S = \{1, 2, 3, 5, 8, 13, 21, 34\}$$. Find the sum $$\sum \max(A)$$ where the sum is taken over all 28 two-element subsets $$A$$ of $$S$$.

 2. (28 p.) A bug moves around a triangle wire. At each vertex it has 1/2 chance of moving towards each of the other two vertices. The probability that after crawling along 10 edges it reaches its starting point can be expressed as $$p/q$$ for positive relatively prime integers $$p$$ and $$q$$. Determine $$p+q$$.

 3. (28 p.) A frog is jumping in the coordinate plane according to the following rules: (i) From any lattice point $$(a,b)$$, the frog can jump to $$(a+1,b)$$, $$(a,b+1)$$, or $$(a+1,b+1)$$. (ii) There are no right angle turns in the frog’s path. How many different paths can the frog take from $$(0,0)$$ to $$(5,5)$$?

 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. (24 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 $$p+q$$.

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