Formula Sheet: Permutations & Combinations | Mathematics For JEE

 Page 1


Page # 28
PERMUTA TION & COMBINNA TION
1. Arrangement : number of permutations of n different things taken r at a
time =
n
P
r
 = n (n ? 1) (n ? 2)... (n ? r + 1) =
)! r n (
! n
?
2. Circular Permutation :
The number of circular permutations of n different things taken all at a
time is; (n – 1)!
3. Selection : Number of combinations of n different things taken r at a
time = 
n
C
r
 =
)! r n ( ! r
! n
?
 =
! r
P
r
n
4. The number of permutations of 'n' things, taken all at a time, when 'p' of
them are similar & of one type, q of them are similar & of another type, 'r' of
them are similar & of a third type & the remaining n ? (p + q + r) are all
different is 
! r ! q ! p
! n
.
5. Selection of one or more objects
(a) Number of ways in which atleast one object be selected out of 'n'
distinct objects is
n
C
1
 + 
n
C
2
 + 
n
C
3
 +...............+ 
n
C
n
 = 2
n
 – 1
(b) Number of ways in which atleast one object may be selected out
of 'p' alike objects of one type  'q' alike objects of second type and
'r' alike of third type is
(p + 1) (q + 1) (r + 1) – 1
(c) Number of ways in which atleast one object may be selected
from 'n' objects where 'p' alike of  one type 'q' alike of second type
and 'r' alike of third type and rest
n – (p + q + r) are different, is
(p + 1) (q + 1) (r + 1) 2
n – (p + q + r)
 – 1
6. Multinomial Theorem :
Coefficient of x
r
 in expansion of (1 ? x)
?n
 = 
n+r ?1
C
r
 (n ? N)
7. Let N = p
a.
 q
b.
 r
c.
..... where p, q, r...... are distinct primes & a, b, c..... are
natural numbers then :
(a) The total numbers of divisors of N including 1 & N is
= (a + 1) (b + 1) (c + 1)........
(b) The sum of these divisors is =
(p
0 
+ p
1 
+ p
2 
+.... + p
a
) (q
0 
+ q
1 
+ q
2 
+.... + q
b
) (r
0 
+ r
1 
+ r
2 
+.... + r
c
)........
Page 2


Page # 28
PERMUTA TION & COMBINNA TION
1. Arrangement : number of permutations of n different things taken r at a
time =
n
P
r
 = n (n ? 1) (n ? 2)... (n ? r + 1) =
)! r n (
! n
?
2. Circular Permutation :
The number of circular permutations of n different things taken all at a
time is; (n – 1)!
3. Selection : Number of combinations of n different things taken r at a
time = 
n
C
r
 =
)! r n ( ! r
! n
?
 =
! r
P
r
n
4. The number of permutations of 'n' things, taken all at a time, when 'p' of
them are similar & of one type, q of them are similar & of another type, 'r' of
them are similar & of a third type & the remaining n ? (p + q + r) are all
different is 
! r ! q ! p
! n
.
5. Selection of one or more objects
(a) Number of ways in which atleast one object be selected out of 'n'
distinct objects is
n
C
1
 + 
n
C
2
 + 
n
C
3
 +...............+ 
n
C
n
 = 2
n
 – 1
(b) Number of ways in which atleast one object may be selected out
of 'p' alike objects of one type  'q' alike objects of second type and
'r' alike of third type is
(p + 1) (q + 1) (r + 1) – 1
(c) Number of ways in which atleast one object may be selected
from 'n' objects where 'p' alike of  one type 'q' alike of second type
and 'r' alike of third type and rest
n – (p + q + r) are different, is
(p + 1) (q + 1) (r + 1) 2
n – (p + q + r)
 – 1
6. Multinomial Theorem :
Coefficient of x
r
 in expansion of (1 ? x)
?n
 = 
n+r ?1
C
r
 (n ? N)
7. Let N = p
a.
 q
b.
 r
c.
..... where p, q, r...... are distinct primes & a, b, c..... are
natural numbers then :
(a) The total numbers of divisors of N including 1 & N is
= (a + 1) (b + 1) (c + 1)........
(b) The sum of these divisors is =
(p
0 
+ p
1 
+ p
2 
+.... + p
a
) (q
0 
+ q
1 
+ q
2 
+.... + q
b
) (r
0 
+ r
1 
+ r
2 
+.... + r
c
)........
Page # 29
 be resolved as a product of two
factors is
=  
? ? square perfect a is N if 1 .... ) 1 c ( ) 1 b ( ) 1 a (
square perfect a not is N if .... ) 1 c ( ) 1 b ( ) 1 a (
2
1
2
1
? ???
???
(d) Number of ways in which a composite number N can be resolved
into two factors which are relatively prime (or coprime) to each
other is equal to 2
n ?1
 where n is the number of different prime
factors in N.
8. Dearrangement :
Number of ways in which 'n' letters can be put in 'n' corresponding
envelopes such that no letter goes to correct envelope is n
 
!
?
?
?
?
?
?
? ? ? ? ? ?
! n
1
) 1 ( .. ..........
! 4
1
! 3
1
! 2
1
! 1
1
1
n