Combinatorics Formula Sheet
Factorial
n!  n(n  1)(n  2)
0!  1
factorial of a non-negative integer n :
special case :
3 2 1
Combinations
n Different Objects Taken r Objects at a Time :
n
n
n!
Cr    
r ! (n  r )!
r
Permutations
n Different Objects :
n!
n Different Objects Taken r Objects at a Time :
n Objects Not All Different (Distinguishable P’s) :
n Different Objects arranged in a Circle :
n!
(n  r )!
(2) n Pr  n(n  1)(n  2)
n!
n1 ! n2 ! n3 !
nk !
(n  1)!
(1)
n
Pr 
(n  r  1)
Binomial Expansion
1.
2.
3.
4.
5.
6.
The number of the terms in the expansion of (a  b)n is n  1 .
The coefficient of the first term is 1.
The coefficient of any other term is the product of the coefficient of the preceding term
and the exponent of a in the preceding term divided by the number of the preceding term.
The exponent of a in any term after the first term is one less than the exponent
of a in the preceding term. (The powers of a decrease from n to 0.)
The exponent of b in any term after the first term is one greater than the exponent
of b in the preceding term. (The powers of b increase from 0 to n .)
The sum of the exponents of a and b in each term is n .
BW – R0214
Binomial Theorem
n
n
n
n
(a  b)n    a n    a n  1 b1    a n  2 b2    a n  3 b3 
0
1
 2
 3
n
   bn
 n
Alternate Form :
( a  b)
n

n

r0
n n  r r
b
 a
r
n
where   
r
n
Cr 
n!
r ! (n  r )!
k -th term Formula :
 n  n  ( k  1) k  1
k-th term of (a  b)n is 
b
a
 k  1
Row
Pascal’s Triangle
0
1
1
2
1
1
3
1
4
5
6
7
1
1
1
1
7
2
3
4
5
6
1
3
6
10
15
21
1
4
10
20
35
1
1
5
15
35
1
6
21
The Inclusion-Exclusion Principle
For any two sets A and B , n ( A  B)  n ( A)  n ( B)  n ( A  B) .
The Complement Principle
If set A is a subset of a universal set U , then n ( A )  n (U )  n ( A C ) .
1
7
1