1.1 Basic Cases and Circular Permutations In the last lecture we considered the basic concepts of permutations and combinations. Let’s consider a couple of examples further to ease the process of understanding these. Ex -1 Find the number of permutations of the word ‘TRIANGLE’. How many of these permutations start with ‘T’ and end with… Continue reading
Week – 10 | Combinatorics – 1
1.1 Introduction Combinatorics, in a way, deals with the art of counting, and thus, a lot of problems in this regard will be dealing with counting the number of arrangements of a particular sequence. As trivial as it may sound, this branch of mathematics has been widely used in various fields like economics, physics and… Continue reading
Week – 10 | Inequalities – 4
Q-1) For the minimum value of is __ Q-2) If be the sides of a triangle, then the minimum value of is a) b) c) d) Q-3) For positive reals , is always a) b) c) d) Q-4) For positive reals in H.P which of the follwing… Continue reading
INMO – 1986 | A Problem on Polynomials
Q) If is a polynomial with integer coefficients and three distinct integers, then show that it is impossible to have Solution: The solution revolves around an important property of integer coefficient Polynomials, For any 2 integers , is always divisible by . So from this we have divides which implies divides , divides , divides . Once we have this, the remaining… Continue reading
Week -9 | Inequalities – 3
Q-1) Define with for . Show that for all positive integers Solution: Using the AM-HM Inequality we have The RHS on simplification becomes . This is known as the famous Nesbitt’s Inequality Q-2) Find the maximum value of with . Solution: Set and . We thus have , and we need the maximum value of . Applying the… Continue reading