Let n be a positive integer. Then, the continued product of first n natural numbers is called factorial n.
it is denoted by `n!` .

Thus , `n! = n (n-1 ) (n-2 ) ......... 3 * 2 * 1`

e.g. `5! = 5 xx 4 xx 3 xx 2 xx 1 = 120` and `4! = 4 xx 3 xx 2 xx 1 = 24`

1. `0! = 1! = 1`

2. Factorials of negative integers and fractions are not defined .

3. `n! = n (n-1)! = n (n-1) ( n-2)!`

4. `(n!)/( r!) = n (n-1) (n-2) ............(r+1)!`

Each of different arrangements which can be made by taking some or all of a number of things, is
called a permutation and number of permutations of n distinct objects taking rat a time is denoted by

`text()^(n)P_r`.

`text()^P_r = (n!)/( (n-r)! ) , AA 0 le r le n = n (n-1) (n-2) .........(n-r +1), AA n in N ` and `r in W`

• The number of permutations of n distinct objects taken
all at a time is `text()^(n)P_n = n!`

• ` text()^(n)P_0 =1, text()^(n)P_1 =n` and `text()^(n)P_(n-1) = n!`

• `text()^(n)P_r = n text()^(n-1)P_(r-1) = r^(n-1) P_(r-1) + text()^(n-1)P_r`

• `text()^(n-1)P_r = (n-r)^(n-1) P_(r-1)`

(i) Arranging n distinct objects, taken r at a time
equivalent to filling r places from n things

Number of arrangements= Number of ways of filling
r places

`= n (n-1)(n-2) ...........(n-r +1 )`

`= ( n (n-1 ) (n-2) .............(n-r +1) { ( n-r)! } )/( ( n-r )! )`

`= (n! )/( (n-r)! ) = text()^(n)P_r`

(ii) The number of arrangements of n different objects
taken all at a time

`= text()^(n)P_n = n!`

(a) `text()^(n)P_0 = (n!)/(n!) =1 , text()^(n)P_r = n^(n-1) P_(r-1)`

(b) `0! =1 ,1/((-r)!) =0` or ` ( (-r)! ) = oo` `[ :. r in N]`

(i) The number of permutations (arrangements) of n different objects, taken r at a time, when each
object may occur once, twice, thrice, ... upto r times in any arrangement is equal to the number of ways of
filling r places, where each place can be filled by anyone of n objects.

Number of permutations= Number of ways of filling

r places `= (n )r`

(ii) The number of arrangements that can be formed using n objects out of which p are identical (and of
one kind), q are identical (and of another kind), r are identical (and of another kind) and the rest are

distinct is `(n!)/( p! q! r!)`

If objects are arranged along a closed curve, then
permutation is known as circular permutation.

(i) The number of circular permutations of n different things taken all at a time is `(n - 1 )!` . If clockwise and
anti-clockwise orders are taken as different.

(ii) If clockwise and anti -clockwise circular
permutations are considered to be the same, then it
is `( (n-1)! )/2`

(iii) Number of circular permutations of n different
things, taken r at a time, when clockwise and
anti-clockwise orders are taken as different is `( text()^(n)P_r)/r` .

(iv) Number of circular permutations of n different
things, taken r at a time, when clockwise and
anti-clockwise orders are not different, is ` ( text()^(n)P_r)/(2r)`.

(v) Number of circular permutations of n things, when p
are alike and the rest different, taken all at a time
distinguishing clockwise and anti-clockwise
arrangements is `( (n-1)! )/( p!)`

Each of the different groups or selections which can be formed by taking some or all of the number of objects,
irrespective of their arrangements, is called a combination.

If `text()^(n)C_r` , denotes the number of combinations of n different

things taken r at a time, then `text()^(n)C_r = (n!)/(r! (n-r)! ) = ( text()^(n)P_r )/(r!) `, where

`r < n , n in N` and `r in W`

• ` text()^(n)C_r = text()^(n)C_(n-r)`

• ` text()^(n)C_r + text()^(n)C_(r-1) = text()^(n+1)C_r`

• ` text()^(n)C_r = 0 ` , if `r not in {1,2,3,............., n }`

• ` text()^(n)P_r = text()^(n)C_r * r!`

(i) Number of ways in which atleast one object is
selected out of n distinct objects

`=> text()^(n)C_1 + text()^(n)C_2 + text()^(n)C_3 +........+ text()^(n)C_n = 2^n -1`

(ii) The total number of ways in which it is possible to make groups by taking some or all out of
`n = (n_1 + n_2 + ... ) ` things, when `n_1` are alike (of one kind), `n_2` are alike (of second kind) and so on, is
`{ (n_1 + 1) (n_2 + 1) ... } -1`.

(iii) The number of selections of r objects out of n
identical objects is 1.

(iv) Total number of selections of zero or more objects
from n identical objects is n + 1.

(v) The number of selections taking atleast one out of
`a_1 + a_2 + a_3 + ... + a_n + k` objects, where `a_1` are alike
(of one kind), `a_2` are alike (of second kind) and so on
... `a_n` are alike (of nth kind) and his distinct, is

`[(a_1 +1)(a_2 +1)(a_3 +1) ... (a_n +1)] 2^k -1`.

(vi) If `N = p^(a_1) * q^(a_2) * r^(a_3) ...............,` are natural numbers , then

(a) Total number of divisors of N (including 1 and N)

`= (a_1 + 1) (a_2 + 1) (a_3 + 1) ...`

(b) Total num.ber of divisors of N (excluding 1 and N)

`= {(a_1 + 1) (a_2 + 1) (a_3 + 1 ) ... } -2`

(c) Total number of divisors of N (excluding 1 or N)

`= { (a_1 +1) (a_2 +1) (a_3 +1) ......... } -1`

(d) The sum of these divisors is

`= (p^0 + p^1 + p^2 + ......+ p^(a_1)) ( q^0 + q^1 + q^2 + .......+ q^(a_2)) ( r^0 + r^1 + r^2 +..........+ r^(a_3) )`

(e) The number of ways in which a composite
number N can be decomposed into two factors is

`text ( Case I )` When N is not a perfect square, then

`text ( Case II )` When N is a perfect square, then

`1/2 [ { (a_1 +1 ) ( a_2 +1 ) ( a_3 +1 ) ......... } +1 ]`

(f) The number of ways in which a composite number N can be decomposed in two-two
factors which are coprime, is ' 2^(k-1)` , where k is the
number of different factors of N.

The functional and the geometrical applications of
permutations and combinations are as given below:

1 . The number of all permutations (arrangements) of n
different objects taken rat a time,

(i) when a particular object is always included in
each arrangement, is ` text()^(n-1)C_(r-1) xx (r!)`

(ii) when a particular object is never taken in each
arrangement , is ` text()^(n-1)C_r xx (r!)`

2. If the sets A has m elements and B has n elements, then

(i) the number of functions from A to B is `n^m`.

(ii) the number of one-one functions from A. to B is
`text()^(n)P_m ` where `m le n`

(i) Number of triangles formed from n points, when no
three points are collinear, is `text()^(n)C_3`.

(ii) Out of n non-concurrent and non-parallel straight
lines, the points of intersection are `text()^(n)C_2`.

(iii) Number of parallelograms in two systems of parallel
lines (when 1st system contains m parallel lines and

lind system containsn paralle lines ) `= text()^(n)C_2 xx text()^(m)C_2`·

(iv) The number of diagonals in a polygon of n sides, is

`text()^(n)C_2 -n`.

(v) The number of total triangles formed by joining the
n points on a plane of which m are collinear, is

`text()^(n)C_3 - text()^(m)C_3 `.

(vi) The number of total different straight lines formed
by joining the n points on a plane of which m are
collinear, is ` text()^(n)C_2 - text()^(m)C_2 + 1`

(vii ) The number of rectangles of any size in a square of

`n xx n` is `sum_(r=1)^n r^3` and number of squares of any size is `sum_(r=1)^n r^2` .

Any integer greater than l can be expressed as product of primes

e.g. `7 = 7^1 , 12 = 2^2 * 3^1 , 360 = 2^3 * 3^2 * 5^1`

Let `n = p_(1)^(alpha_1) * p_(2)^(alpha_2) * P_3^(alpha_3) * ... * p_r^(alph_r)` , where `P_i, i = 1, 2, ... ,r` are
distinct primes and `alpha_i, i = 1 , 2, ... , r` are positive integers.

(i) Number of divisors of n is

` (alpha_1 + 1) (alpha_2 +1) (alpha_3 + 1) ........ (alpha_r +1)`

(ii) Sum of divisors of n is

` (P_(1)^(alpha_3 +1) -1 )/( p_1 -1) ( P_(2)^(alpha_2 +1) -1 )/(P_2 -1)............( P_(r)^(alpha_r +1) -1)/(P_r -1)`

(iii) lf p is a prime and p' divides, then `r = [ n/p ] + [ n/p^2 ] + ............`

The division of objects into groups are taken place as when
the objects are different and identical as given below:

1 . The number of ways of dividing n different objects into
3 groups of p,q and `r (p + q + r = n)` is

(i) `(n!)/( p! q! r!)` ; p,q and r are unequal.

(ii) `(n!)/(p! 2! (q!)^2 ) ; q = r `

(iii) `(n! )/( 3! (p!)^3 )` p =q=r


2. The number of·ways of dividing n different objects into
r groups is


`1/(r!) [ r^n - ( ( r,1) ) (r-1)^n + ( ( r,2 ) ) (r-2)^n - ( ( r,3 ) ) (r-3)^n + ...... ]`


3. The number of ways of dividing n different objects into
r groups taking into account the order of the groups
and also the order of objects in each group, is


`text()^(n+r-1 )P_n = r (r+1) (r+2) ..........( r+n -1 )`.

1. The number of ways of dividing n identical objects among r persons, such that each gets 1,2,3, ... objects,
is the coefficient of `x^(n-r)` in the expansion of
`(1+ x+ x^2 + ......+ x^(k-1) )^r` .

2. The number of ways of dividing n identical objects among r person such that each one may get atmost
n objects , is ` ( (n+r -1 ) , ( r-1 ) )`
In other words, the total number of ways of dividing n
identical objects into r groups, if blank groups are
allowed , is `text()^(n+r -1)C_(r-1)`

3. The total number of ways of dividing n identical objects
among r persons and each one of them, receives atleast
one item , is `text()^(n-1)C_(r-1)`.

In other words, the number of ways in which n
identical things can be divided into r groups such that
blank groups are not allowed, is `text()^(n-1)C_(r-1)`.

4. The total number of selections of some or all out of
p + q + r items, where p are alike of one kind, q are
alike of second kind and rest are alike of third kind, is
`[(p + 1)-(q + 1) * (r + 1 ) -1 ]`.

1. The number of ways in which r objects can be
selected from n different objects, if k particular
objects are

(a) always included `= text()^(n-k)C_(r-k)`

(b) never include `=text()^(n-k)C_r`


2. The number of arrangements of n distinct objects
taken rat a time, so that!? particular objects are

(a) always included= `text()^(n-k)C_(r-k) * r!`

(b) never include `= text()^(n-k)C_(r ) * r!`

Any change in the given order of the things, is called a
dearrangement.

1. If n things form an arrangement in a row, then the number of ways in which they can be dearranged, so
that none of them occupies its original place, is

`n! (1- 1/(1!) +1/(2!) -1/(3!) +.......+ (-1)^n * 1/(n!) )`.

2. If n things arc arranged at n places, then the number
of ways to rearrange exactly r things at right places, is

`(n!)/(r!) [ 1- 1/(1!) + 1/(2!) -1/(3!) + 1/(4!) +......+ (-1)^(n-r) 1/( (n-r)! ) ]`