Applied Discrete Structures by Alan Doerr, Kenneth Levasseur

By Alan Doerr, Kenneth Levasseur

Textbook from UMass Lowell, model 3.0

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Position k can be filled by any one of n (k 1) = n k + 1 elements 30 CHAPTER 2. COMBINATORICS Hence, by the rule of products, P (n, k) = n · (n 1) · (n 2) · · · · · (n k + 1) = n! (n k)! It is important to note that the derivation of the permutation formula given above was done solely through the rule of products. This serves to reiterate our introductory remarks in this section that permutation problems are really rule-ofproducts problems. We close this section with several examples. 9 (Another example of choosing officers).

How many five-card poker hands using 52 cards contain exactly two aces? 10. In poker, a full house is three-of-a-kind and a pair in one hand; for example, three fives and two queens. How many full houses are possible from a 52-card deck? You can use the sage cell in the Sage Note to do this calculation, but also write your answer in terms of binomial coefficients. 11. A class of twelve computer science students are to be divided into three groups of 3, 4, and 5 students to work on a project. How many ways can this be done if every student is to be in exactly one group?

7. If a positive integer is a multiple of 100, we can identify this fact from its decimal representation, since it will end with two zeros. What can you say about a positive integer if its binary representation ends with two zeros? What if i ends in k zeros? 8. Can a multiple of ten be easily identified from its binary representation? 5 Summation Notation and Generalizations Most operations such as addition of numbers are introduced as binary operations. That is, we are taught that two numbers may be added together to give us a single number.

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