lecturenotes_4_5 - Combinatorics: The Fine Art of Counting...

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Combinatorics: The Fine Art of Counting Lecture Notes - Weeks 4 and 5 – Binomial Coefficients Note – to improve the readability of these lecture notes, we will assume that multiplication takes precedence over division, i.e. A / B*C always means A / (B*C). Binomial coefficients are written horizontally, i.e. (4 2) means “4 choose 2” or 4!/2!2! Introduction In the previous lecture we looked at several different counting examples that involved the binomial coefficient (n 2). These examples fell into 4 major categories: 1. Counting Subsets – people shaking hands, edges in a graph, and unordered pairs are all examples of counting subsets of size 2. The answer we found in each case was (n 2) . 2. Summing a Sequence – adding up the numbers from 1 to n or counting the points in a triangle with n points on each side both involve summing an increasing sequence. The answer we got was (n+1 2) . 3. Distinct Partitions – bouquets of flowers of 3 colors and ordered triples (x,y,z) that sum to n were both examples of processes that involved partitioning or categorizing a set of identical objects (e.g. blank flowers, 1s) into distinct groups (e.g. red/white/pink, x/y/z). We used 2 separators to divide the objects into 3 groups and then chose the position of the two separators from among n+2 possible positions for objects and separators, obtaining (n+2 2) . 4. Block Walking – we counted the number of direct routes along an grid of city blocks 2 blocks north/south by n blocks east/west (note that each block in the grid is bounded by streets so there are 3 parallel streets running east/west and n+1 parallel streets running north/south) . Each direct route was exactly n+2 blocks long and we had to choose 2 points along our route to walk north, resulting in (n+2 2) . We will see that all these counting problems generalize in a straight-forward manner, allowing us to replace 2 with k in each of the cases above. We will eventually see one more example, the Binomial Theorem, which is the origin of the mysterious name “binomial coefficients”. Counting Subsets While there are many ways to define the binomial coefficient (n k), counting subsets can be regarded as the most fundamental. This is why we say (n k) or “n choose k” means the number of ways of choosing a subset of k elements from a set with n elements (as opposed to defining (n k) in terms of some algebraic formula). Many of the most basic facts about binomial coefficients follow immediately when we use this definition as our starting point. Recall that every subset of {x 1 , x 2 , …, x n } elements corresponds to a binary string of length n where the i th bit is 1 if and only if x i is in the subset. (n k) = (n n-k)
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This note was uploaded on 01/25/2010 for the course MATH 21127 taught by Professor Howard during the Spring '08 term at Carnegie Mellon.

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lecturenotes_4_5 - Combinatorics: The Fine Art of Counting...

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