# Combinatorics

 1. (10 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 $$n$$.

 2. (24 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 $$\frac{P\cdot 10^{10}}{24^4}$$.

 3. (24 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 $$n$$?

 4. (14 p.) Given a convex polyhedron with 26 vertices, 60 edges and 36 faces, 24 of the faces are triangular and 12 are quadrilaterals. A space diagonal is a line segment connecting two vertices which do not belong to the same face. How many space diagonals does the polyhedron have?

 5. (26 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 $$(5,5)$$?

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