Q) Show that there cannot be distinct positive integers such that and . Solution: This problem trivially arrives to a scenario that we have covered in the module ‘System of Equations’. And it touches a very important concept that is useful in many scenarios. We have . Cubing both sides, and cancelling out the common terms (assuming… Continue reading
Week – 12 | Combinatorics – 6
Q-1) Find the number of triangles whose angular points are at the angular points of a given polygon of n sides, but none of whose sides are on the sides of the given polygon. Solution: The polygon has n sides – hence n angular points. We can choose a triangle from these n angular points… Continue reading
Week -12 | Combinatorics – 5
1.1 Principle of Inclusion and Exclusion This very important principle is a generalization of the Sum Rule to sets which need not be disjoint. Let’s say that we have 2 sets & . We look at the cardinality of the union of these 2 sets (We assume that students going through this module are familiar with… Continue reading
Week – 11 | Combinatorics – 4
1.1 Some Corner Cases We’ve almost covered the essence of combinatorics that are needed at a beginner’s level to start studying this beautiful topic. This will be the last editorial of this module for those of you who’d not like to pursue any advanced studies in this chapter. There’ll be another editorial on this topic… Continue reading
Week – 11 | Combinatorics – 3
1.1 Introduction to Generating Functions Consider a simple problem where we have to calculate the number of ways to choose 2 fruits from 5 distinct fruits. Let’s call them A,B,C,D,E. So we have 1 each of these 5 fruits and we need to choose 2 of them (ignoring the order). A simple way would be… Continue reading