Counting (2nd Edition) by Koh Khee Meng, Tay Eng Guan

By Koh Khee Meng, Tay Eng Guan

Publication Description
Publication Date: January 25, 2013
This e-book in its moment version is an invaluable, beautiful creation to easy counting options for top secondary to undergraduate scholars, in addition to academics. more youthful scholars and lay those who savour arithmetic, let alone avid puzzle solvers, also will locate the ebook attention-grabbing. many of the difficulties and functions listed here are strong for increase talent in counting. also they are necessary for honing easy talents and strategies regularly challenge fixing. a number of the difficulties keep away from regimen and the diligent reader will usually become aware of multiple means of fixing a specific challenge, that's certainly an enormous knowledge in challenge fixing. The booklet therefore is helping to provide scholars an early begin to studying problem-solving heuristics and considering skills.

New chapters initially from a supplementary booklet were additional during this version to considerably elevate the assurance of counting thoughts. the hot chapters comprise the primary of Inclusion and Exclusion, the Pigeonhole precept, Recurrence family members, the Stirling Numbers and the Catalan Numbers. a few new difficulties have additionally been extra to this edition.

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How many shortest P—Q routes in the grid are there? Solution Certainly, we can solve the problem directly by listing all the possible shortest routes. This, however, would not be practical if we wish to solve the same problem in, say, a 190 x 100 rectangular grid. We look for a more efficient way. There are two types of segments: horizontal and vertical. Let us use a "0" to represent a horizontal segment, and a " 1 " to represent a vertical segment. 6 can accordingly be represented by the binary sequence with four "0"s and two "l"s as shown below: -> 010100 P •- Likewise, we can have: _J -> 000101 and so on.

3 shows five distinct points on the circumference of a circle. How many chords can be formed by these points? Let A be the set of such chords, and B, the set of 2-element subsets of {1,2,3,4,5}. 4). Then / is a mapping from A to B. )• Thus, / is injective. 4 The Bijection Principle 39 q = 5), there is a chord a in A (in this instance, a is the chord joining points 2 and 5) such that f(a) = {p,q}. Thus, / is onto. Hence, / : A ->• B is a bijection and, by (BP), we have \A\ = \B\. As 5 is the set of all 2-element subsets of {1,2,3,4,5}, \B\ = g ) .

While a "combination" of S is just a subset of S (and thus the ordering of elements is immaterial), a "permutation" of S is an arrangement of certain elements of S (and thus the ordering of elements is important). 2 r£) = (n-r + l ) ^ ) , where r > 1. +1 = P? _1; (ii) P r n + 1 = r! + r{P^ + P^1 + ••• + P;_ x ); (iii) (n - r)P? 3 Prove that the product of any n consecutive integers is divisible by n!. 4 Find the sum l - l ! + 2-2! + 3-3! + --- + n - n ! This page is intentionally left blank Chapter 4 Applications Having introduced the concepts of r-permutations and r-combinations of an n-element set, and having derived the formulae for P?

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