Practice Problems Q-1) There are p intermediate stations on a railway line between 2 points A and B. In how many ways can a train stop at 3 of the stations such that no 2 of the stopping stations are consecutive? Q-2) How many different numbers smaller than can be formed by the digits 0,1,2… Continue reading
Category: Algebra
Week – 12 | Combinatorics – 7
This is an optional module for students preparing for the engineering entrance exams. Q-1) Among 6 persons in a room, there are either 3 who know each other or 3 who are complete strangers Solution : Let us consider a hexagon with each person denoting a specific vertex of the hexagon. We join 2 vertices… Continue reading
INMO 1988 | A problem on Equations
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