Mathematics_Today_-_October_2016

FUNDAMENTAL PRINCIPLES OF COUNTING
z Multiplication Principle
If an operation A can be performed in ‘m’ different
ways and a second operation B can be performed in
‘n’ different ways and C is a work which is done only
when both A and B are done, then the number of ways
of doing the work C is m × n. This can be extended to
any finite number of operations.
If there are n jobs J 1 , J 2 , ..., Jn such that job Ji can be
performed independently in mi ways ; i = 1, 2, ...., n and
there is work C which is done only when all the works
(J 1 , J 2 , ..., Jn) are done. Then the number of ways of doing
the work C is m 1 × m 2 × m 3 ×........× mn.
z Addition Principle
If an operation A can be performed in ‘m’ different ways
and another operation B, which is independent of the
first operation, can be performed in ‘n’ ways and C is
a work which is done only when either A or B is done,
then the number of ways of doing the work C is (m + n),
this can be extended to any infinite number of mutually
exclusive operations.
PERMUTATIONS
Each of the different arrangements which can be made
by taking some or all of a number of distinct objects at
a time is called a permutation. Permutation of things
means arrangement of things. The word arrangement is
used if order of things is taken into account. Thus if order
of different things changes, then their arrangement also
changes.
Theorems
z The number of permutations of n different things,
taken r at a time (repetition not allowed) is denoted

This article is a collection of shortcut methods, important formulas and MCQs along with their detailed solutions which provides
an extra edge to the readers who are preparing for various competitive exams like JEE(Main & Advanced) and other PETs.

PERMUTATIONS & COMBINATIONS

by nPr, where nPr is defined as n

nr

!

()!−

.

z The number of permutations of n different things

taken all at a time (repetition not allowed) = n!.

z The number of permutations of n different things,

taken r at a time when each thing may be repeated

any number of times is nr.

z Total number of arrangements of the n objects

taken all at a time when each thing may be repeated

any number of times is nn.

z The number of permutations of n things taken all

at a time where p are alike of one kind, q are alike

of second kind, r are alike of third kind and rest all

are different is n

pqr

!

!!!

.

COMBINATIONS

Each of the different groups or selections which can

be made by taking some or all of a number of distinct

objects or items, irrespective of their arrangements or

order in which they are placed, is called a combination.

Combination means selection only and permutation

means selection + arrangement.

Theorem : The number of combinations of n different

things, taken r at a time (repetition not allowed) is

denoted by nCr where nCr = − ≤≤

n

rn r

! rn

!( )!

,. 0

CONDITIONAL COMBINATION

z Number of ways of choosing r things out of n given

things if p particular things must be excluded is

(n – p)Cr.

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