Permutations and Combinations
From LocusWiki
Welcome to the chapter on Permutations and Combinations
Brief Introduction about the Chapter
This chapter is one of the most interesting chapters that we’ll study at this level. The beauty and challenge of this branch of mathematics lies in the innumerous tricks and mathematical artifices that abound in this subject. Clarity of thought, more than any thing else, is what is required to understand the subject properly. Also, you’d do best if you refrain from memorising any formulae or particular cases here; concentrate on building a logical approach, solving everything from first principles.
The main objective of this chapter is to count. Given a set of things or objects or persons ( or whatever), we need to arrange a subset
of
(according to some constraints) or select a subset
of
(again, according to certain criteria). In fact, we are actually interested in counting the number of such arrangements or selections. Read the following questions:
“From a team of 15 cricket players, how can we select a playing team of 11players?
“There are 20 people whom we need to seat in 2 rows of 10 seats each. How many ways exist of doing so?
“From a deck of 52 playing cards, in how many ways can we select two red cards and three black cards?”
“How many rectangles exist on a standard 8 × 8 chessboard?”
“How many factors does 144000 have? In general, how many factors does a natural number have?”
These are some of the many types of questions which we’ll learn to solve in this chapter.
We’ll build a systematic approach to deal with such counting issues. To really appreciate the beauty of the solving techniques that we’ll develop, you are urged to try out each and every question that we solve here, on your own first, and only then look at the solution. Only this approach will help you solve counting questions elegantly.
