5.4 Binomial Coefficients
• Theorem 1: The binomial theorem Let x and y be
variables, and let n be a nonnegative integer. Then
 n  n j j
( x  y )     x y
j 0  j 
 n  n  n  n 1
 n  n2 2
 n 
n n
n 1
 x y    y
   x    x y    x y    
0
1
 2
 n  1
n
n
n
• Example 3: What is the coefficient of x12y13 in the
expansion of (x + y)25?
1
The Binomial Theorem
• Corollary 1: Let n be a nonnegative integer. Then
n
n



2

k 
k 0  
n
• Corollary 2: Let n be a positive integer. Then
n
(1)    0

k 0
k 
n
k
• Corollary 3: Let n be a nonnegative integer. Then
n
n


(
2
)

3

k 
k 0
 
n
k
2
Pascal’s Identity and Triangle
• Theorem 2: Pascal’s Identity Let n and k be positive
integers with n  k . Then
 n  1  n   n 

  
   
 k   k  1  k 
3
Pascal’s Identity and Triangle
FIGURE 1 Pascal’s Triangle.
4
Some Other Identities of the Binomial
Coefficients
• Theorem 3: Vandermonde’s Identity Let m, n, and r
be nonnegative integers with r not exceeding either
m or n. Then  m  n  r  m  n 

 r
   
 
 k 0  r  k  k 
• Corollary 4: If n is nonnegative integer, then
 2n 
n
     
 n  k 0  k 
n
2
• Theorem 4: Let n and r be nonnegative integers with
r  n. Then
n
n 1
j


 

    
 r  1 j  r  r 
5