1. Relationships Between
Binomial Coefficients

2. Relationships Between
Binomial Coefficients
Binomial Theorem

3. Relationships Between
Binomial Coefficients
Binomial Theorem   

n
k
k
k
nn
xCx
0
1

4. Relationships Between
Binomial Coefficients
Binomial Theorem   

n
k
k
k
nn
xCx
0
1
n
n
nk
k
nnnn
xCxCxCxCC  2
210

5. Relationships Between
Binomial Coefficients
Binomial Theorem   

n
k
k
k
nn
xCx
0
1
n
n
nk
k
nnnn
xCxCxCxCC  2
210
e.g. (i) Find the values of;

n
k
k
n
C
1
a)

6. Relationships Between
Binomial Coefficients
Binomial Theorem   

n
k
k
k
nn
xCx
0
1
n
n
nk
k
nnnn
xCxCxCxCC  2
210
e.g. (i) Find the values of;

n
k
k
n
C
1
a)   

n
k
k
k
nn
xCx
0
1

7. Relationships Between
Binomial Coefficients
Binomial Theorem   

n
k
k
k
nn
xCx
0
1
n
n
nk
k
nnnn
xCxCxCxCC  2
210
e.g. (i) Find the values of;

n
k
k
n
C
1
a)
let x = 1;
  

n
k
k
k
nn
xCx
0
1
  

n
k
k
k
nn
C
0
111

8. Relationships Between
Binomial Coefficients
Binomial Theorem   

n
k
k
k
nn
xCx
0
1
n
n
nk
k
nnnn
xCxCxCxCC  2
210
e.g. (i) Find the values of;

n
k
k
n
C
1
a)
let x = 1;
  

n
k
k
k
nn
xCx
0
1
  

n
k
k
k
nn
C
0
111


n
k
k
nn
C
0
2

9. Relationships Between
Binomial Coefficients
Binomial Theorem   

n
k
k
k
nn
xCx
0
1
n
n
nk
k
nnnn
xCxCxCxCC  2
210
e.g. (i) Find the values of;

n
k
k
n
C
1
a)
let x = 1;
  

n
k
k
k
nn
xCx
0
1
  

n
k
k
k
nn
C
0
111


n
k
k
nn
C
0
2


n
k
k
nnn
CC
1
02

10. Relationships Between
Binomial Coefficients
Binomial Theorem   

n
k
k
k
nn
xCx
0
1
n
n
nk
k
nnnn
xCxCxCxCC  2
210
e.g. (i) Find the values of;

n
k
k
n
C
1
a)
let x = 1;
  

n
k
k
k
nn
xCx
0
1
  

n
k
k
k
nn
C
0
111


n
k
k
nn
C
0
2


n
k
k
nnn
CC
1
02
0
1
2 CC nn
n
k
k
n


11. Relationships Between
Binomial Coefficients
Binomial Theorem   

n
k
k
k
nn
xCx
0
1
n
n
nk
k
nnnn
xCxCxCxCC  2
210
e.g. (i) Find the values of;

n
k
k
n
C
1
a)
let x = 1;
  

n
k
k
k
nn
xCx
0
1
  

n
k
k
k
nn
C
0
111


n
k
k
nn
C
0
2


n
k
k
nnn
CC
1
02
0
1
2 CC nn
n
k
k
n

12
1

n
n
k
k
n
C

12.  7531b) CCCC nnnn

13.  7531b) CCCC nnnn
  

n
k
k
k
nn
xCx
0
1

14.  7531b) CCCC nnnn
  

n
k
k
k
nn
xCx
0
1
 5
5
4
4
3
3
2
210 xCxCxCxCxCC nnnnnn

15.  7531b) CCCC nnnn
let x = 1;
  

n
k
k
k
nn
xCx
0
1
 5
5
4
4
3
3
2
210 xCxCxCxCxCC nnnnnn
   54321011 CCCCCC nnnnnnn

16.  7531b) CCCC nnnn
let x = 1;
  

n
k
k
k
nn
xCx
0
1
 5
5
4
4
3
3
2
210 xCxCxCxCxCC nnnnnn
   54321011 CCCCCC nnnnnnn
 12 543210  CCCCCC nnnnnnn

17.  7531b) CCCC nnnn
let x = 1;
  

n
k
k
k
nn
xCx
0
1
 5
5
4
4
3
3
2
210 xCxCxCxCxCC nnnnnn
   54321011 CCCCCC nnnnnnn
 12 543210  CCCCCC nnnnnnn
let x = -1;    54321011 CCCCCC nnnnnnn

18.  7531b) CCCC nnnn
let x = 1;
  

n
k
k
k
nn
xCx
0
1
 5
5
4
4
3
3
2
210 xCxCxCxCxCC nnnnnn
   54321011 CCCCCC nnnnnnn
 12 543210  CCCCCC nnnnnnn
let x = -1;    54321011 CCCCCC nnnnnnn
 20 543210  CCCCCC nnnnnn

19.  7531b) CCCC nnnn
let x = 1;
  

n
k
k
k
nn
xCx
0
1
 5
5
4
4
3
3
2
210 xCxCxCxCxCC nnnnnn
   54321011 CCCCCC nnnnnnn
 12 543210  CCCCCC nnnnnnn
let x = -1;    54321011 CCCCCC nnnnnnn
 20 543210  CCCCCC nnnnnn
subtract (2) from (1)

20.  7531b) CCCC nnnn
let x = 1;
  

n
k
k
k
nn
xCx
0
1
 5
5
4
4
3
3
2
210 xCxCxCxCxCC nnnnnn
   54321011 CCCCCC nnnnnnn
 12 543210  CCCCCC nnnnnnn
let x = -1;    54321011 CCCCCC nnnnnnn
 20 543210  CCCCCC nnnnnn
subtract (2) from (1)
 531 2222 CCC nnnn

21.  7531b) CCCC nnnn
let x = 1;
  

n
k
k
k
nn
xCx
0
1
 5
5
4
4
3
3
2
210 xCxCxCxCxCC nnnnnn
   54321011 CCCCCC nnnnnnn
 12 543210  CCCCCC nnnnnnn
let x = -1;    54321011 CCCCCC nnnnnnn
 20 543210  CCCCCC nnnnnn
subtract (2) from (1)
 531 2222 CCC nnnn

531
1
2 CCC nnnn

22. 
n
k
k
n
Ck
1
c)

23. 
n
k
k
n
Ck
1
c)
  

n
k
k
k
nn
xCx
0
1

24. 
n
k
k
n
Ck
1
c)
Differentiate both sides
  

n
k
k
k
nn
xCx
0
1

25. 
n
k
k
n
Ck
1
c)
Differentiate both sides
  

n
k
k
k
nn
xCx
0
1
  1
0
1
1 


 k
n
k
k
nn
xCkxn

26. 
n
k
k
n
Ck
1
c)
Differentiate both sides
  

n
k
k
k
nn
xCx
0
1
  1
0
1
1 


 k
n
k
k
nn
xCkxn
let x = 1;   


n
k
k
nn
Ckn
0
1
11

27. 
n
k
k
n
Ck
1
c)
Differentiate both sides
  

n
k
k
k
nn
xCx
0
1
  1
0
1
1 


 k
n
k
k
nn
xCkxn
let x = 1;   


n
k
k
nn
Ckn
0
1
11
    


n
k
k
nnn
CkCn
1
0
1
02

28. 
n
k
k
n
Ck
1
c)
Differentiate both sides
  

n
k
k
k
nn
xCx
0
1
  1
0
1
1 


 k
n
k
k
nn
xCkxn
let x = 1;   


n
k
k
nn
Ckn
0
1
11
    


n
k
k
nnn
CkCn
1
0
1
02
  1
1
2



n
n
k
k
n
nCk

29.  
 
n
k
k
nk
k
C
0 1
1
d)

30.  
 
n
k
k
nk
k
C
0 1
1
d)
  

n
k
k
k
nn
xCx
0
1

31.  
 
n
k
k
nk
k
C
0 1
1
d)
Integrate both sides
  

n
k
k
k
nn
xCx
0
1

32.  
 
n
k
k
nk
k
C
0 1
1
d)
Integrate both sides
  

n
k
k
k
nn
xCx
0
1
 
11
1 1
0
1



 


 k
x
CK
n
x kn
k
k
n
n

33.  
 
n
k
k
nk
k
C
0 1
1
d)
Integrate both sides
  

n
k
k
k
nn
xCx
0
1
 
11
1 1
0
1



 


 k
x
CK
n
x kn
k
k
n
n
let x = 0;
 
1
0
1
01 1
0
1



 


 k
CK
n
kn
k
k
n
n

34.  
 
n
k
k
nk
k
C
0 1
1
d)
Integrate both sides
  

n
k
k
k
nn
xCx
0
1
 
11
1 1
0
1



 


 k
x
CK
n
x kn
k
k
n
n
let x = 0;
 
1
0
1
01 1
0
1



 


 k
CK
n
kn
k
k
n
n
1
1



n
K

35.  
 
n
k
k
nk
k
C
0 1
1
d)
Integrate both sides
  

n
k
k
k
nn
xCx
0
1
 
11
1 1
0
1



 


 k
x
CK
n
x kn
k
k
n
n
let x = 0;
 
1
0
1
01 1
0
1



 


 k
CK
n
kn
k
k
n
n
1
1



n
K
let x = -1;    
1
1
1
111
1
0
1








 k
C
n
kn
k
k
n
n

36.  
 
n
k
k
nk
k
C
0 1
1
d)
Integrate both sides
  

n
k
k
k
nn
xCx
0
1
 
11
1 1
0
1



 


 k
x
CK
n
x kn
k
k
n
n
let x = 0;
 
1
0
1
01 1
0
1



 


 k
CK
n
kn
k
k
n
n
1
1



n
K
let x = -1;    
1
1
1
111
1
0
1








 k
C
n
kn
k
k
n
n
 
1
1
1
1
1
0 






 nk
C
kn
k
k
n

37.  
 
n
k
k
nk
k
C
0 1
1
d)
Integrate both sides
  

n
k
k
k
nn
xCx
0
1
 
11
1 1
0
1



 


 k
x
CK
n
x kn
k
k
n
n
let x = 0;
 
1
0
1
01 1
0
1



 


 k
CK
n
kn
k
k
n
n
1
1



n
K
let x = -1;    
1
1
1
111
1
0
1








 k
C
n
kn
k
k
n
n
 
1
1
1
1
1
0 






 nk
C
kn
k
k
n
 
1
1
1
1
0 



 nk
C
kn
k
k
n

38.  
      nnn
n
xxx
xii
2
111
identity;theofsidesbothonoftscoefficientheequatingBy

 
 2
2
0 !
!2
that;show
n
n
k
nn
k








39.  
      nnn
n
xxx
xii
2
111
identity;theofsidesbothonoftscoefficientheequatingBy

 
 2
2
0 !
!2
that;show
n
n
k
nn
k







  

n
k
k
k
nn
xCx
0
1

40.  
      nnn
n
xxx
xii
2
111
identity;theofsidesbothonoftscoefficientheequatingBy

 
 2
2
0 !
!2
that;show
n
n
k
nn
k







  

n
k
k
k
nn
xCx
0
1
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC  



1
1
2
2
2
210 

41.  
      nnn
n
xxx
xii
2
111
identity;theofsidesbothonoftscoefficientheequatingBy

 
 2
2
0 !
!2
that;show
n
n
k
nn
k







  

n
k
k
k
nn
xCx
0
1
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC  



1
1
2
2
2
210 
   nnn
xxx  11inoftcoefficien

42.  
      nnn
n
xxx
xii
2
111
identity;theofsidesbothonoftscoefficientheequatingBy

 
 2
2
0 !
!2
that;show
n
n
k
nn
k







  

n
k
k
k
nn
xCx
0
1
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC  



1
1
2
2
2
210 
   nnn
xxx  11inoftcoefficien
 
 n
n
nn
n
nn
n
nnnn
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC
xCxCxCxCxCC










1
1
2
2
2
210
1
1
2
2
2
210



43.  
      nnn
n
xxx
xii
2
111
identity;theofsidesbothonoftscoefficientheequatingBy

 
 2
2
0 !
!2
that;show
n
n
k
nn
k







  

n
k
k
k
nn
xCx
0
1
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC  



1
1
2
2
2
210 
   nnn
xxx  11inoftcoefficien
n
x
n
nn













0






































 
01122
122 n
x
n
n
x
n
x
n
n
x
n
x
n
n nnn

 
 n
n
nn
n
nn
n
nnnn
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC
xCxCxCxCxCC










1
1
2
2
2
210
1
1
2
2
2
210



44.  
      nnn
n
xxx
xii
2
111
identity;theofsidesbothonoftscoefficientheequatingBy

 
 2
2
0 !
!2
that;show
n
n
k
nn
k







  

n
k
k
k
nn
xCx
0
1
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC  



1
1
2
2
2
210 
   nnn
xxx  11inoftcoefficien
n
x
n
nn













0
1
11














 n
x
n
n
x
n
 
 n
n
nn
n
nn
n
nnnn
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC
xCxCxCxCxCC










1
1
2
2
2
210
1
1
2
2
2
210



45.  
      nnn
n
xxx
xii
2
111
identity;theofsidesbothonoftscoefficientheequatingBy

 
 2
2
0 !
!2
that;show
n
n
k
nn
k







  

n
k
k
k
nn
xCx
0
1
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC  



1
1
2
2
2
210 
   nnn
xxx  11inoftcoefficien
n
x
n
nn













0
1
11














 n
x
n
n
x
n 22
22














 n
x
n
n
x
n
 
 n
n
nn
n
nn
n
nnnn
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC
xCxCxCxCxCC










1
1
2
2
2
210
1
1
2
2
2
210



46.  
      nnn
n
xxx
xii
2
111
identity;theofsidesbothonoftscoefficientheequatingBy

 
 2
2
0 !
!2
that;show
n
n
k
nn
k







  

n
k
k
k
nn
xCx
0
1
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC  



1
1
2
2
2
210 
   nnn
xxx  11inoftcoefficien
n
x
n
nn













0
1
11














 n
x
n
n
x
n 22
22














 n
x
n
n
x
n






































 
01122
122 n
x
n
n
x
n
x
n
n
x
n
x
n
n nnn

 
 n
n
nn
n
nn
n
nnnn
n
n
nn
n
nn
n
nnnn
xCxCxCxCxCC
xCxCxCxCxCC










1
1
2
2
2
210
1
1
2
2
2
210



47. 


















































022110
oftcoefficien
n
n
n
n
nn
n
nn
n
nn
xn


48. 












kn
n
k
n
But



















































022110
oftcoefficien
n
n
n
n
nn
n
nn
n
nn
xn


49. 












kn
n
k
n
But
2222
210

























n
nnnn




















































022110
oftcoefficien
n
n
n
n
nn
n
nn
n
nn
xn


50. 












kn
n
k
n
But
2222
210

























n
nnnn









n
k k
n
0
2



















































022110
oftcoefficien
n
n
n
n
nn
n
nn
n
nn
xn


51. 












kn
n
k
n
But
2222
210

























n
nnnn









n
k k
n
0
2
  nn
xx
2
1inoftcoefficien 



















































022110
oftcoefficien
n
n
n
n
nn
n
nn
n
nn
xn


52. 












kn
n
k
n
But
2222
210

























n
nnnn









n
k k
n
0
2
  nn
xx
2
1inoftcoefficien 
  nnn
x
n
n
x
n
n
x
n
x
nn
x 222
2
22
2
2
1
2
0
2
1 





























 



















































022110
oftcoefficien
n
n
n
n
nn
n
nn
n
nn
xn


53. 












kn
n
k
n
But
2222
210

























n
nnnn









n
k k
n
0
2
  nn
xx
2
1inoftcoefficien 
  nnn
x
n
n
x
n
n
x
n
x
nn
x 222
2
22
2
2
1
2
0
2
1 





























 



















































022110
oftcoefficien
n
n
n
n
nn
n
nn
n
nn
xn


54. 












kn
n
k
n
But
2222
210

























n
nnnn









n
k k
n
0
2
  nn
xx
2
1inoftcoefficien 
  nnn
x
n
n
x
n
n
x
n
x
nn
x 222
2
22
2
2
1
2
0
2
1 





























 



















































022110
oftcoefficien
n
n
n
n
nn
n
nn
n
nn
xn








n
n
xn 2
oftcoefficien

55. Now
      nnn
xxx
2
111 

56. Now
      nnn
xxx
2
111 












 n
n
k
nn
k
2
0
2

57. Now
      nnn
xxx
2
111 












 n
n
k
nn
k
2
0
2
 
!!
!2
nn
n


58. Now
      nnn
xxx
2
111 












 n
n
k
nn
k
2
0
2
 
!!
!2
nn
n

 
 2
!
!2
n
n


59. Now
      nnn
xxx
2
111 












 n
n
k
nn
k
2
0
2
 
!!
!2
nn
n

 
 2
!
!2
n
n

Exercise 5F;
4, 5, 6, 8, 10,15
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