SUMMARY
⇒ Fundamental principles of counting
(a) Multiplication Rule: If certain thing may be done in ‘m’ different ways and when
it has been done, a second thing can be done in ‘n ‘ different ways then total number
of ways of doing both things simultaneously = m × n.
(b) Addition Rule: It there are two alternative jobs which can be done in ‘m’ ways
and in ‘n’ ways respectively then either of two jobs can be done in (m + n) ways.
⇒ Factorial: The factorial n, written as n! or n , represents the product of all integers from 1 to n both inclusive. To make the notation meaningful, when n = o, we define o! or o
= 1.
Thus, n! = n (n – 1) (n – 2) ….. …3.2.1
⇒ Permutations: The ways of arranging or selecting smaller or equal number of persons
or objects from a group of persons or collection of objects with due regard being paid to
the order of arrangement or selection, are called permutations.
The number of permutations of n things chosen r at a time is given by
nPr =n ( n – 1 ) ( n – 2 ) … ( n – r + 1 )
where the product has exactly r factors.
⇒ Circular Permutations: (a) n ordinary permutations equal one circular permutation.
Hence there are nPn/ n ways in which all the n things can be arranged in a circle. This
equals (n–1)!.
(b) the number of necklaces formed with n beads of different colours
⇒ Combinations: The number of ways in which smaller or equal number of things are
arranged or selected from a collection of things where the order of selection or
arrangement is not important, are called combinations.
(a) nCr has a meaning only when r and n are integers 0≤ r ≤ n and nCn–r has a meaning only when 0 ≤ n – r ≤ n.
Permutations when some of the things are alike, taken all at a time
Permutations when each thing may be repeated once, twice,…upto r times in any
arrangement = n!.
⇒ The total number of ways in which it is possible to form groups by taking some or all of
n things (2n –1).
The total, number of ways in which it is possible to make groups by taking some or all
out of n (=n1 + n2 + n3 +…) things, where n1 things are alike of one kind and so on, is
given by { (n1 + 1) ( n2 + 1) ( n3 + 1)…} –1
⇒ The combinations of selecting r1 things from a set having n1 objects and r2 things from a set having n2 objects where combination of r1 things, r2 things are independent is given
by