![]() Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$ The Rule of Product − If a sequence of tasks $T_1, T_2, \dots, T_m$ can be done in $w_1, w_2, \dots w_m$ ways respectively and every task arrives after the occurrence of the previous task, then there are $w_1 \times w_2 \times \dots \times w_m$ ways to perform the tasks. ![]() $A \cap B = \emptyset$), then mathematically $|A \cup B| = |A| + |B|$ If we consider two tasks A and B which are disjoint (i.e. The Rule of Sum − If a sequence of tasks $T_1, T_2, \dots, T_m$ can be done in $w_1, w_2, \dots w_m$ ways respectively (the condition is that no tasks can be performed simultaneously), then the number of ways to do one of these tasks is $w_1 + w_2 + \dots +w_m$. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. ![]() Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. For solving these problems, mathematical theory of counting are used. For instance, in how many ways can a panel of judges comprising of 6 men and 4 women be chosen from among 50 men and 38 women? How many different 10 lettered PAN numbers can be generated such that the first five letters are capital alphabets, the next four are digits and the last is again a capital letter. Then, we discuss what the running time of merge sort would be.In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. We describe, informally, how we can divide and merge the subproblems, yielding a recurrence relation for the runtime of merge sort. We want to determine an unknown linear order on, and we want to do this by dividing the problem into different subproblems. ![]() Ultimately, the lower bound is simplified by using Stirling’s approximation. Then, a theorem is discussed which gives a lower bound on the length of time an arbitrary sorting algorithm can take in the worst-case scenario. This video introduces the sorting problem, and gives an example of a poor sorting algorithm. We ask a set of questions about this list that leads us back to the fair division problem. Suppose, for example, we have a list of n distinct positive integers. In this video we start to develop a framework for understanding the difficulty of a problem. This video introduces big-oh and little-oh notation, and provides a few examples that use these concepts. It is often difficult to say, precisely, what the running times for an algorithm might be. In this video, we consider a list of increasing functions, and ask how quickly they tend to infinity. Within the framework of a mathematics course, we introduce the idea of a problem size and the concept of running time. Here is an easy application of the Pigeon Hole Principle. The Erdős-Szekeres Theorem is introduced, and a proof of this theorem is provided that uses the Pigeon Hole Principle. In this video, Professor Trotter explains the Erdős number, and tells some stories about this famous mathematician. ![]() This short video introduces the Pigeon Hole Principle, as well as a generalization of it. The Multinomial Theorem gives us an expansion when the base has more than two terms, like in (x 1+x 2+x 3) n. The Binomial Theorem gives us as an expansion of (x+y) n. Here we introduce the Binomial and Multinomial Theorems and see how they are used. How many ways can you rearrange the letters of a string if some of the letters are duplicated? The answer is given by multinomial coefficients. You may want to download the lecture slides that were used for these videos (PDF). ![]()
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