Chapter 6: Permutations and Combinations

How many ways can $5$ books be arranged on a shelf? How many committees of $4$ can be formed from $10$ people? How many license plates can be made with $3$ letters and $4$ digits? Combinatorics , the art of counting , gives precise answers to all such questions, often without ever listing the possibilities.

This chapter introduces two complementary ideas. A permutation is an arrangement where order matters (the first book, then the second book, then the third). A combination is a selection where order does not matter (a committee of four). The same group of people forms many different "arrangements in a queue" but exactly one "committee".

Both ideas rest on a single principle , the fundamental principle of counting (multiplication principle). From it we derive the factorial $n! = n(n-1)\cdots 1$ , the permutation count $P(n, r) = \dfrac{n!}{(n-r)!}$ , and the combination count $\binom{n}{r} = \dfrac{n!}{r!(n-r)!}$ .

For Class XII and JEE you will use these formulas in:

Binomial theorem (next chapter): coefficients are $\binom{n}{r}$ .
Probability: counting favourable outcomes.
Algebra: counting solutions to equations.
Discrete optimisation: counting routes, colourings, codes.

The single biggest test in this chapter is modelling: can you read a word problem and decide whether order matters (permutation), whether repetition is allowed, and what the right formula is?

What's inside

Fundamental principle of counting , the multiplication and addition principles.
Factorials , definition, computation, simplifications.
Permutations of distinct objects , $P(n, r)$ and arrangements.
Permutations with repetition and identical objects , circular and special cases.
Combinations , $\binom{n}{r}$ and selections.
Mixed and applied problems , words, committees, geometry, hand-shakes.

Key results / Formula card

Concept	Formula
Multiplication principle	If $A$ has $m$ outcomes and $B$ has $n$ , then " $A$ then $B$ " has $m \cdot n$
Factorial	$n! = n(n-1)(n-2)\cdots 1$ , $0! = 1$
Permutation $n$ distinct, $r$ at a time	$P(n, r) = \dfrac{n!}{(n-r)!}$
Permutations of $n$ with repetition allowed	$n^r$ ( $r$ positions)
Permutations of $n$ objects, $p$ of one kind, $q$ of another	$\dfrac{n!}{p!\,q!\,\cdots}$
Circular permutations of $n$ distinct	$(n-1)!$
Combination $n$ distinct, $r$ at a time	$\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$
Symmetry	$\binom{n}{r} = \binom{n}{n - r}$
Pascal's identity	$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$
$\binom{n}{0} = \binom{n}{n} = 1$

How to read this chapter

For every problem ask three questions: (i) Does order matter? (ii) Is repetition allowed? (iii) Are all the objects distinct? Your answer to these chooses the formula. Once you have it, the computation is mechanical.

Chapter 6: Permutations and Combinations

What's inside

Key results / Formula card

How to read this chapter

Sub-topics

Practice quiz