\(\renewcommand{\d}{\displaystyle} 1 Sets and Lists 2 Binomial Coefcients 3 Equivalence Relations Homework Assignments 4 1 Sets and Lists WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed. Simple is harder to achieve. /Length 1235 \newcommand{\va}[1]{\vtx{above}{#1}} xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d You can use all your notes, calcu-lator, and any books you Axiom 1 Every probability is between 0 and 1 included, i.e: Axiom 2 The probability that at least one of the elementary events in the entire sample space will occur is 1, i.e: Axiom 3 For any sequence of mutually exclusive events $E_1, , E_n$, we have: Permutation A permutation is an arrangement of $r$ objects from a pool of $n$ objects, in a given order. on Introduction. ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream a b. of the domain. x[yhuv*Nff&oepDV_~jyL?wi8:HFp6p|haN3~&/v3Nxf(bI0D0(54t,q(o2f:Ng #dC'~846]ui=o~{nW] 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. \newcommand{\isom}{\cong} Problem 2 In how many ways can the letters of the word 'READER' be arranged? Axioms of probability For each event $E$, we denote $P(E)$ as the probability of event $E$ occurring. Complemented Lattice : Every element has complement17. / [(a_1!(a_2!) A Set is an unordered collection of objects, known as elements or members of the set.An element a belong to a set A can be written as a ∈ A, a A denotes that a is not an element of the set A. A permutation is an arrangement of some elements in which order matters. x3T0 BCKs=S\.t;!THcYYX endstream Then, The binomial expansion using Combinatorial symbols. << Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). /Length 1781 Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. It includes the enumeration or counting of objects having certain properties. Let s = q + r and s = e f be written in lowest terms. of irreflexive relations = 2n(n-1), 15. 'A`zH9sOoH=%()+[|%+&w0L1UhqIiU\|IwVzTFGMrRH3xRH`zQAzz`l#FSGFY'PS$'IYxu^v87(|q?rJ("?u1#*vID =HA`miNDKH;8&.2_LcVfgsIVAxx$A,t([k9QR$jmOX#Q=s'0z>SUxH-5OPuVq+"a;F} Last Minute Notes Discrete Mathematics - GeeksforGeeks { (k-1)!(n-k)! } From his home X he has to first reach Y and then Y to Z. How many ways can you distribute \(10\) girl scout cookies to \(7\) boy scouts? Define the set Ento be the set of binary strings with n bits that have an even number of 1's. in the word 'READER'. If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. Prove or disprove the following two statements. >> endobj We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. Cram sheet/Cheat sheet/study sheet for a discrete math class that covers sequences, recursive formulas, summation, logic, sets, power sets, functions, combinatorics, arrays and matrices. Bnis the set of binary strings with n bits. /Width 156 Expected value The expected value of a random variable, also known as the mean value or the first moment, is often noted $E[X]$ or $\mu$ and is the value that we would obtain by averaging the results of the experiment infinitely many times.
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