Combinatorial Identities Angie R. and Girish V.

Combinatorics “the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties” from Wolfram MathWorld How many ways are there to make a three-layer ice-cream cone if we have an unlimited number of vanilla, chocolate, and strawberry scoops? How many diagonals are in an octagon? How many ways are there to choose a random group of six officers from Mu Alpha Theta?

Basic definitions We call this “n choose k” The number of ways to choose k objects from n things if order does not matter (for example, choosing a committee, not choosing president and vice president) What is 8 choose 3? What is 9 choose 4? What is 8 choose 5? What is 9 choose 5?

Combinatorial identities

More Examples!

Sums in Pascal’s triangle -Very useful summation formula: -Proof: Count the number of subsets of {1,2,...,n} in two ways.

Problem -JZ is an odd boy and likes to play with subsets. He picks a random subset of {1,2,...,9}. What is the probability that his subset has size that is odd?

Solution/hints -What are the subsets with odd sizes? -Well, they are 1,3,5,7,9 -Now how many are there and can we use a previous identity to find the answer?

Problem -JZ has reformed his ways and now is less odd. He can now tolerate even numbers, but will not tolerate large numbers. JZ chooses a random subset of {1,2,...,9}. What is the probability that the size of the subset is less than 5?

Solution/hints -Just like last time -Sizes of subsets can be 0,1,2,3,4 -Use same identity to find the answer

Vandermonde’s Identity -Complicated way of asking that given m+n people, how many ways are there to choose a committee of r people? -Useful when the number of things being chosen is constant

Problem -Suppose that there are two boxes, one with 4 balls and one with 5 balls. A total of 4 balls are to be chosen so that some of them are from the first box and the rest of them are from the second box. How many ways can this be done?

Solution/Hints -Try using casework -Once the cases are established, just use Vandermonde

Problem -Given an integer k, compute the value of

Solution/Hints -How will we apply Vandermonde to this? -Is there a previous identity we can use?

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