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Discrete MathematicsA grocery store employee is stocking apples. Each apple is a different color. There are 10 apples left in the box and the employee pulls out 2 of them at random. What is the probability that the employee pulls out one pink apple and yellow apple?20237mDiscrete MathematicsWhat is pigeonhole principle? Using it, prove the following: (a) During a month with 30 days, a baseball team plays at least one game a day, but no more than 45 games. Show that there must be a period of some number of consecutive days during which the team must play exactly 14 games.20227mDiscrete MathematicsProve the following by using the principle of mathematical induction for all n \in N: 1^3 + 2^3 + 3^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2202314mDiscrete MathematicsIf m is the number of objects (pigeons) and n is the number of boxes (pigeonholes), then the function is both one-to-one and onto if (i) m<n (ii) m=n (iii) m>n (iv) none of these20232m
PreviousWrite short notes on the following: (i) Forward proof (ii) Disjunctive and conjunctive normal form (iii) Fundamental theorem of arithmetic This question covers topics from different modules: * (i) Forward proof: This falls under Module 3: Propositional Logic & Proof Techniques. * (ii) Disjunctive and conjunctive normal form: This also falls under Module 3: Propositional Logic & Proof Techniques, as these are concepts in propositional logic. * (iii) Fundamental theorem of arithmetic: This is a concept from number theory, which is typically covered in Module 2: Mathematical Induction & Counting Techniques (as number theory is often grouped with induction and counting in discrete mathematics courses).NextProve Bernoulli's inequality that if h > -1, then 1+nh \le (1+h)^n for all non-negative integers n.
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