Problem Set 8. Combinatorics

# Problem Set 8. Combinatorics - Problem Seminar Fall 2013...

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Problem Seminar. Fall 2013. Problem Set 8. Combinatorics. Classical results. 1. The equation x 1 + x 2 + . . . + x r = n has exactly ( n + r - 1 r - 1 ) non-negative integer solutions. 2. Consider a convex polygon with n vertices so that no 3 diagonals go through the same point. (a) How many intersection points do the diagonals have? (b) Into how many regions do the diagonals divide the interior of the polygon? 3. An unfair coin (probability p of showing heads) is tossed n times. What is the probability that the number of heads will be even? 4. Catalan numbers. Find a closed-form expression for the number of valid sequences containing n pairs of parentheses. For example, when n = 2 , there are 2 valid sequences: ()() and (()) . The sequence ())( is not valid. Problems. 1. Put 92. B1. Let S be a set of n distinct real numbers. Let A S be the set of numbers that occur as averages of two distinct elements of S . For a given n 2 , what is the smallest possible number of distinct elements in A S ?

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