Combinatorics

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By Bradley Jackson, Dmitri Thoro

Publication via Jackson, Bradley, Thoro, Dmitri

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The binary number system is the system for representing numbers using the base b, where b = 2. In this system the only possible digits are 0 and 1. This base is widely used in computers since the choice of a 0 or a 1 can be indicated by the presence or absence of an electrical current flowing in a given circuit. Binary numbers are also useful in combinatorics. The following theorem shows that it is always possible to represent an integer using any base b, where b > 1. THEOREM I If b is any integer greater than 1, every integer a> 0 can be uniquely represented in the form a = 0, + + ...

7 x A Pair of Dice Example 3 A die is a cube with 1, 2, 3, 4, 5, 6 dots, respectively, on each of its six faces. In various games of chance a pair of dice are rolled. If the two dice are different, say, one is colored green and the other is colored red, then how many different outcomes are possible when the two dice are rolled? Figure I . I. 1 I I I5 Solution Think of recording each outcome as an ordered pair, the green die in the first coordinate and the red die in the second coordinate. Since there are six outcomes for each die, the total number of outcomes equals the total number of ordered pairs where each digit is an element of {l,2,3,4,5,6}.

In that case, 0 <(n + 1) — b = when r>0, a=(—q—1)b+(b—r). Ifb>r>0, then b>b—r>O and the division theorem holds for all integers. ) When analyzing a problem involving an integer it is often helpful to express the integer as a product of primes. 2 (Fundamental Theorem of Arithmetic) Every integer n> 1 can be expressed as n = Pi P2 Pm, where each p1 is a prime. Proof If n is itself a prime, the result is obvious and in particular the result holds for n = 2. Assume that the result is true for all 2 k n.

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