Chapter I: Permutations, combinations, occupancy problems. We start with some basic counting principles and examples. I.1. Two ways of counting the same ﬁnite set give the same answer. Example I.2. (Hand shaking lemma). The number of delegates at a conference who shake hands an odd number of times is even. Proof. Let D1 , ..., Dn be the delegates, and let X = {(i, j) : Di and Dj shake hands}, and let k = |X|. We count k in two ways. First k is even because it is twice the number of handshakes. Secondly k = k1 + ... + kn where ki = the number of times that Di shakes hands. THUS k1 + .. + kn is even, which implies that evenly many of the ki are odd, which is what we wanted to prove. I.3. (Multiplication principle.) If we make k successive choices, and for 1 ≤ i ≤ k the number of ways of making the ith choice given the previous choices is ni . THEN total number of choices is n1 × ... × nk . I.4. For k ≤ n the number of ways of arranging (or permuting, or selecting in order) k out of n objects, is n × (n − 1) × .. × (n − k + 1) = n!/(n − k)!. This number is also denoted P (n, k). Example I.5. In a race with 20 horses the number of ways in which the ﬁrst 3 places can be ﬁlled is 20 × 19 × 18 = P (20, 3) = 6840. I.6. Note that P (n, n) = n! = the number of permutations of a set of n objects. I.7. The number of ways of choosing or selecting k out of n objects (without regard to order) is denoted C(n, k) (C stands for “combination”). This is 1

the same as the number of k element subsets of an n-element set. Moreover C(n, k) = P (n, k)/k! = n!/(n − k)!k!. Proof. Each choice of k out of n objects gives rise to k! permutations of k out of n objects. So C(n, k).k! = P (n, k). ETC. Note C(n, k) = C(n, n − k). (Why? Because to choose k out of n is the same thing as choosing the complement.) Example I.8. The number of poker hands drawn from a normal pack of cards is C(52, 5) = P (52, 5)/5!. I.9. The number of subsets...

...LECTURENOTES
OPS 5095 ONLINE FORMAT
WEEK ONE
Welcome to class!
In today's complex and highly competitive business environment, it is critical for you to understand the importance of service operations. In a service business, operations is truly where the "rubber meets the road".
OPS 5095 is an intensive course in which you will learn strategic and tactical views of service operations, as well as gain valuable knowledge about...

...ECON 1023 Fall 2011
Instructor: Gibson Nene
Chapter 1LectureNotes: Limits, Alternatives and Choices
The economic perspective or economic way of thinking takes the following concepts into consideration:
* Scarcity and Choice
* Purposeful Behavior
* Marginalism: Benefits and Costs
Scarcity and Choice
Economics is about wants and means:
* Society has the resources to make goods and services that satisfy our many desires....

...OT310 Genesis/Van Wick
Lecture #1
Section Covered: Genesis 1
The word Genesis in Greek means origin. The Greek Septuagint translation of this is bereshith, or beginnings. The authorship of the book Genesis is Moses, as found in Romans 10 19 But I ask, did Israel not understand? First Moses says,
“I will make you jealous of those who are not a nation; with a foolish nation I will make you angry.” 1 Further evidence that Moses...

...BIOB11- Lecture1 Part 1
First of all, just a quick review: there is quite a bit of difference between the prokaryotic and eukaryotic cells. They have different internal structures. Eukaryotic cells have internal membrane structures like nucleus, ER, Golgi; whereas they do not exist in the prokaryotic cells. When you look at the genetic material, DNA, you can see that it’s just present in the cytoplasm of the prokaryotic cell while in the eukaryotic...

...Geology, Atmospheric Science and Geography) to the study of the environment, and the solution of environmental problems. Environmental science provides an integrated, quantitative, and interdisciplinary approach to the study of environmental systems.[1]
Related areas of study include environmental studies and environmental engineering. Environmental studies incorporates more of the social sciences for understanding human relationships, perceptions and policies towards the...

...Week One Assignment
Chapter One
The Sociological Perspective
The sociological perspective provides a specific view of the world. Briefly describe this perspective and the assumptions this perspective is based on.
The sociological perspective puts emphasis on the social context in which people live in and how these contexts influence their lives. At the center of this perspective is the question of how groups influence people, especially how people are influence by their...

...Week1 Discussion HRM595
DQ1
Given this scenario, how have Linda and Larry handled the conflict thus far? What should they do now?
If Linda and Larry have already asked Dale and Donna about limiting the noise personally then they should tell the apartment manager. In alerting management they can ask what course of action will be taken along with an estimated time frame. If nothing has changed then Linda and Larry can use the last resort in calling the...

...South Asian Religion NotesWeek1
Hindu: Persian term for people living below the Hindus river. It wasn't till the 1850s the term hinduism was used. Hindus would call themselves in relation to their communities and which Gods they held as dominant. There is no shared system of belief and not a single historical point of origin for the name or foundation for the religion as a whole. It encompasses a variet and number of different religious groups...