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. So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$.You may want to download the lecture slides that were used for these videos (PDF). There are $50/6 = 8$ numbers which are multiples of both 2 and 3. There are $50/3 = 16$ numbers which are multiples of 3. How many integers from 1 to 50 are multiples of 2 or 3 but not both?įrom 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. Examplesįrom a set S =|A_1 \cap \dots \cap A_2|$ In other words a Permutation is an ordered Combination of elements. PermutationsĪ permutation is an arrangement of some elements in which order matters. Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). Solution − From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). How many ways are there to go from X to Z? From there, he can either choose 4 bus routes or 5 train routes to reach Z. He may go X to Y by either 3 bus routes or 2 train routes. From his home X he has to first reach Y and then Y to Z. Question − A boy lives at X and wants to go to School at Z. 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. In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events.
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