1. General Expansion of
Binomials

2. General Expansion of
Binomials
 kk
k
n
xxC 1inoftcoefficientheis

3. General Expansion of
Binomials
 kk
k
n
xxC 1inoftcoefficientheis







k
n
Ck
n

4. General Expansion of
Binomials
 kk
k
n
xxC 1inoftcoefficientheis
  n
n
nnnnn
xCxCxCCx  2
2101







k
n
Ck
n

5. General Expansion of
Binomials
 kk
k
n
xxC 1inoftcoefficientheis
  n
n
nnnnn
xCxCxCCx  2
2101
which extends to;
  n
n
nn
n
nnnnnnnn
bCabCbaCbaCaCba  

 1
1
22
2
1
10 







k
n
Ck
n

6. General Expansion of
Binomials
 kk
k
n
xxC 1inoftcoefficientheis
  n
n
nnnnn
xCxCxCCx  2
2101
 4
32.. xge 
which extends to;
  n
n
nn
n
nnnnnnnn
bCabCbaCbaCaCba  

 1
1
22
2
1
10 







k
n
Ck
n

7. General Expansion of
Binomials
 kk
k
n
xxC 1inoftcoefficientheis
  n
n
nnnnn
xCxCxCCx  2
2101
 4
32.. xge 
which extends to;
  n
n
nn
n
nnnnnnnn
bCabCbaCbaCaCba  

 1
1
22
2
1
10 
       4
4
43
3
422
2
43
1
44
0
4
33232322 xCxCxCxCC 







k
n
Ck
n

8. General Expansion of
Binomials
 kk
k
n
xxC 1inoftcoefficientheis
  n
n
nnnnn
xCxCxCCx  2
2101
 4
32.. xge 
which extends to;
  n
n
nn
n
nnnnnnnn
bCabCbaCbaCaCba  

 1
1
22
2
1
10 
       4
4
43
3
422
2
43
1
44
0
4
33232322 xCxCxCxCC 
432
812162169616 xxxx 







k
n
Ck
n

9. Pascal’s Triangle Relationships

10. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n

11. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx

12. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 

13. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 
k
xoftscoefficienatlooking

14. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 
k
xoftscoefficienatlooking
k
n
CLHS

15. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 
k
xoftscoefficienatlooking
k
n
CLHS

16. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 
k
xoftscoefficienatlooking
k
n
CLHS      k
n
k
n
CCRHS 1
1
1
11 




17. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 
k
xoftscoefficienatlooking
k
n
CLHS      k
n
k
n
CCRHS 1
1
1
11 



k
n
k
n
CC 1
1
1 




18. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 
k
xoftscoefficienatlooking
k
n
CLHS      k
n
k
n
CCRHS 1
1
1
11 



k
n
k
n
CC 1
1
1 


 k
n
k
n
k
n
CCC 1
1
1 




19. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 
k
xoftscoefficienatlooking
k
n
CLHS      k
n
k
n
CCRHS 1
1
1
11 



k
n
k
n
CC 1
1
1 


 k
n
k
n
k
n
CCC 1
1
1 



 
l"symmetricaistrianglesPascal'"
11where2   nkCC kn
n
k
n

20. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 
k
xoftscoefficienatlooking
k
n
CLHS      k
n
k
n
CCRHS 1
1
1
11 



k
n
k
n
CC 1
1
1 


 k
n
k
n
k
n
CCC 1
1
1 



 
l"symmetricaistrianglesPascal'"
11where2   nkCC kn
n
k
n
  13 0  n
nn
CC

21. Pascal’s Triangle Relationships
  11where1 1
1
1
 


nkCCC k
n
k
n
k
n
     1
111


nn
xxx
  1
1
111
1
1
1
1
0
1
1 




 n
n
nk
k
nk
k
nnn
xCxCxCxCCx 
k
xoftscoefficienatlooking
k
n
CLHS      k
n
k
n
CCRHS 1
1
1
11 



k
n
k
n
CC 1
1
1 


 k
n
k
n
k
n
CCC 1
1
1 



 
l"symmetricaistrianglesPascal'"
11where2   nkCC kn
n
k
n
  13 0  n
nn
CC
Exercise 5B; 2ace, 5, 6ac,
10ac, 11, 14