z Number of ways of choosing r things out of n
given things if p particular things must be included
(p ≤ r) is n – pCr (^) – (^) p.
CONDITIONAL PERMUTATIONS
z The number of all permutations (arrangements) of
n different objects taken r at a time,
(a) When a particular object is to be always
included in each arrangement is n–1Pr–1 ⋅ r
(b) When a particular object is never taken in
each arrangement is n–1Pr
z When all of a certain given things are not to occur
together :
In order to find the number of permutations
when all of a certain given things are not to occur
together, find
(a) the total number of arrangements when there
is no restriction. Let this number be x.
(b) number of arrangements when all of the
things (which are not to occur together) are
together. Let this number by y.
(c) Required number = x – y.
z When no two of a certain given things occur
together :
In order to find the number of permutations when
no two of a certain given things occur together.
(a) First of all put the m things on which there is
no restriction in a line. These m things can be
arranged in m! ways.
(b) Then count the number of places between
every two of m things on which there is no
restriction including end positions. Number
of such places will be (m + 1).
(c) If m is the number of things on which there is
no restriction and n is the number of things
no two of which are to occur together, then
required number of ways = m + 1Pn × m!.
z If two type of things are to be arranged alternately,
then
(a) if there numbers differ by 1 put the thing
whose number is greater at first, third, fifth ...
places etc. and other things at second, fourth,
sixth .... places. Let the number of numbers be
m + 1 and m then the required number of ways
= (m + 1)! × m!.
(b) If the number of two types of things is same,
consider two cases separately keeping first
type of things at first place, third, fifth place...
etc. and second type of things at first, third,
fifth place ... and then add. Let the number of
things be m, then the required number of ways
= 2 × m! × m!.
Selection of One or More Objects
z Selection from Different Objects
(a) The number of ways of selecting any number
of objects out of n different objects = 2n
(b) The number of ways of selecting atleast one
object out of n different objects
= nC 1 + nC 2 + nC 3 + ... + nCn = 2n – 1
z Selection from Identical Objects
(a) The number of ways of selecting r objects out
of n identical objects is 1.
(b) The number of ways of selecting any number
(zero or more) of objects out of n identical
objects is n + 1.
(c) The total number of selections of some or all
out of x + y + z items where x are alike of one
kind, y are alike of second kind and z are alike
of third kind is (x + 1) (y + 1) (z + 1).
(d) The total number of selections of atleast one
out of x + y + z items where x are alike of one
kind, y are alike of second kind and z are alike
of third kind is [(x + 1) (y + 1) (z + 1)] – 1.
z Selection from Identical and Distinct Objects
(a) If we have x alike objects of one kind, y alike
objects of second kind, z alike objects of third
kind and k different objects, then the number
of ways of selecting any number of objects
= (x + 1) (y + 1) (z + 1). 2 k
(b) If we have x alike objects of one kind, y alike
objects of second kind, z alike objects of
third kind and k different objects, then the
number of ways of selecting atleast one object
= {(x + 1) (y + 1) (z + 1). 2 k} –
DIVISION AND DISTRIBUTION OF DISTINCT
OBJECTS
z The number of ways of dividing n distinct objects in
r groups of different sizes containing a 1 , a 2 , a 3 , ..., ar
objects respectively, where n = a 1 + a 2 + a 3 + ... + ar
and a 1 , a 2 , a 3 , ..., ar are all different numbers
==n ana−−−ana a a a a
r
CC CC n
aa a a
r
1 r
1
2
12
3 ... 123
!
!!!"!
z The number of ways of distributing n distinct objects
among r people such that one of them gets a 1 objects,
some one gets a 2 , some one gets a 3 , ... and some one
gets ar where a 1 + a 2 + ... + ar = n and a 1 , a 2 , a 3 , ..., ar
are different numbers = n
aa a a
r
r
!
!!!!
(^) .!
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