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How to master combinatorics

posted Oct 1, 2011 7:38 AM by Sachin Gadkar

In combinatorics, the most important thing is to understand what are the most important considerations. 
Like the objects on which we are working for different choices we try to find are they distinct or identical. If the objects or choices are done with repetition or without repetition. Moreover, the problem is of ordered choice or unordered choices.
Like for example, if we have 4 indentical marbles and we need to find number of ways of choosing 3 marbles. then there will be only one choice!
How many arrangements (i.e. ordered) of 3 things is ? one

Lets consider another problem, if there are 4 identical objects and we need to find number of ways of choosing 3 objects is C(4,3) and number of arrangements is P(4,3)

In the above given 4 distinct objects if we find how many different arrangements of 3 things are done and this time we have repetition allowed, we have 4x4x4 ways.

SO remember inorder to master this topic you need to learn to solve problems and learn concepts from scratch and not follow any book or the formulae created in that book! Read books but try to evolve those concepts and problems yourself!

Very important relation of root and modulus

posted May 11, 2011 11:09 AM by Sachin Gadkar   [ updated May 18, 2011 10:58 AM ]

We treat the following as one of the ways to define modulus of x 


Difference between Identity and an equation?

posted May 9, 2011 6:34 AM by Sachin Gadkar   [ updated May 9, 2011 7:18 AM ]

An identity is also an equation where the solution set is equal to the Domain of definition of the equation. 
For example, 

Is an identity, since the domain of definition of the equation is R (real numbers) and the values satisfying the equation are all R. 
Lets see another example, 

has the domain of definition of the equation as R- (2n+1) pi/2  and the values of x that satisfy the equation are the same.

While in an equation the solution set is always a subset of the domain of definition

Magic of equivalence in probability

posted Feb 21, 2011 12:15 AM by Sachin Gadkar   [ updated Oct 1, 2011 7:37 AM ]


Irrationals which are transcedental

posted Nov 30, 2010 3:36 AM by Sachin Gadkar   [ updated Jul 3, 2011 11:26 PM ]

I often wonder why the word transcendental is used for irrationals, but mind not all irrationals are transcendental. The word transcendental is used to represent those Irrationals which cannot be expressed as roots of a polynomial with integer coefficients.
e.g. e and pi. 

Now we know this!

Linear differential equation

posted Nov 25, 2010 10:34 PM by Sachin Gadkar

Discontinuity of a function at x=a when f(a) is undefined

posted Nov 18, 2010 1:26 AM by Sachin Gadkar   [ updated Nov 18, 2010 7:22 AM ]

Function is discontinuous at a point x=a  can be of different types. 
Discontinuity in a function is because of function value at x=a is either defined but unequal to LHL or RHL or it doesnt exist.
But actually can we term a function to be discontinuous at x=a if the function is not defined at x=a?
like for example, is it fair to ask this question, Is f(x)=1/x  continuous on Real R? Mathematically, its unfair to ask continuity of a function on space other than its domain. So f(x)=1/x  is continuous on its domain R-{0}  but if forced to answer on R then its discontinuous at x=0 for no mistake (i am personifying the function) of the f(x)

So mathematically, f(a) undefined cannot be imply that f is discontinous at x=a?
For example,   cannot be termed to be discontinuous on its domain, but cannot also be asked to be discontinuous in R.

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