By Sebastian M. Cioaba, M. Ram Murty

The concept that of a graph is prime in arithmetic because it comfortably encodes various kin and enables combinatorial research of many advanced counting difficulties. during this booklet, the authors have traced the origins of graph idea from its humble beginnings of leisure arithmetic to its sleek environment for modeling communique networks as is evidenced via the area broad internet graph utilized by many net se's. This booklet is an creation to graph concept and combinatorial research. it's in line with classes given via the second one writer at Queen's collage at Kingston, Ontario, Canada among 2002 and 2008. The classes have been geared toward scholars of their ultimate yr in their undergraduate program.


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4. We have 7 tasks to do. In how many ways can we choose 5 of them to perform one task a day during 5 consecutive weekdays? 5. How many are there 9-digit natural numbers, containing every digit 1; 2; : : : ; 9 once? 6. How many are there 10-digit natural numbers, containing each digit 0; 1; 2; : : : ; 9 once? 7. How many are there 10-digit numbers with the sum of digits equal 4? 8. Find the sum of all integer numbers containing digits 1; 2; 3; 4, such that any digit occurs in each number once. 9.

If there is the Sum Rule, then it likely should be a Product Rule. To introduce it, we again analyze a model problem. 2. Identification cards on Small Planet contain two characters, one capital Latin letter and one Hindu-Arabic digit, for example, “S 8”. How many are there various cards, if we can use all 26 letters and 10 digits? Solution. First of all, we have to state unequivocally what cards must be considered identical, and what cards are different. Since we consider a mathematical problem, we do not take into consideration size, color, font, etc.

13. How many are there 6-digit odd numbers without repeating digits? How many such numbers begin with a 1? 14. How many permutations of the 10 digits 0; 1; : : : ; 9 contain either the sequence 246 or the sequence 578, but not both? 15. How many permutations of the 10 digits 0; 1; : : : ; 9 contain the sequence 246 or the sequence 680, but not both? 16. How many permutations of the 10 digits contain either the sequence 246, or the sequence 680, or both? 17. How many are there different 10-digit natural numbers consisting only of digits 1, 2, and 3, if a 3 appears precisely two times?

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