CHAPTER 3 Counting
... (a) How many such length-5 lists have at least one letter repeated? (b) How many such length-6 lists have at least one letter repeated? 9. This problem concerns 4-letter codes made from the letters A,B,C,D,...,Z. (a) How many such codes can be made? (b) How many such codes have no two consecutive le ...
... (a) How many such length-5 lists have at least one letter repeated? (b) How many such length-6 lists have at least one letter repeated? 9. This problem concerns 4-letter codes made from the letters A,B,C,D,...,Z. (a) How many such codes can be made? (b) How many such codes have no two consecutive le ...
I CHAPTER 3 Counting
... solution to Part (d) above. It is not hard to see what went wrong. The list (E, E, A, B) is of type 1 and type 2, so it got counted twice. Similarly (E, E, C, E ) is of type 1, 3 and 4, so it got counted three times. In fact, you can find many similar lists that were counted multiple times. In solvi ...
... solution to Part (d) above. It is not hard to see what went wrong. The list (E, E, A, B) is of type 1 and type 2, so it got counted twice. Similarly (E, E, C, E ) is of type 1, 3 and 4, so it got counted three times. In fact, you can find many similar lists that were counted multiple times. In solvi ...
Solutions
... 1 (such numbers are called square-free, for example 7, 15, 21 are square-free, but 9 and 12 are not). A number less than 1000 is divisible by a square of an integer greater than 1 if and only if it is divisible by one of the squared primes 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961(= 312 ) (bec ...
... 1 (such numbers are called square-free, for example 7, 15, 21 are square-free, but 9 and 12 are not). A number less than 1000 is divisible by a square of an integer greater than 1 if and only if it is divisible by one of the squared primes 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961(= 312 ) (bec ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".