The document discusses relationships in Pascal's triangle. It shows that the binomial coefficient nCk can be expressed as n-1Ck-1 + n-1Ck. It also shows that Pascal's triangle is symmetrical, with nCk = nCn-k for 1 ≤ k ≤ n-1.

1. General Expansion of
Binomials

2. General Expansion of
Binomials
Ck is the coefficient of x k in 1  x 
n k

3. General Expansion of
Binomials
Ck is the coefficient of x k in 1  x 
n k
n
n
Ck   
k 

4. General Expansion of
Binomials
Ck is the coefficient of x k in 1  x 
n k
n
n
Ck   
k 
1  x n  nC0  nC1 x  nC2 x 2   nCn x n

5. General Expansion of
Binomials
Ck is the coefficient of x k in 1  x 
n k
n
n
Ck   
k 
1  x n  nC0  nC1 x  nC2 x 2   nCn x n
which extends to;
a  b n  nC0 a n  nC1a n1b nC2 a n2b 2   nCn1ab n1  nCnb n

6. General Expansion of
Binomials
Ck is the coefficient of x k in 1  x 
n k
n
n
Ck   
k 
1  x n  nC0  nC1 x  nC2 x 2   nCn x n
which extends to;
a  b n  nC0 a n  nC1a n1b nC2 a n2b 2   nCn1ab n1  nCnb n
e.g .2  3 x 
4

7. General Expansion of
Binomials
Ck is the coefficient of x k in 1  x 
n k
n
n
Ck   
k 
1  x n  nC0  nC1 x  nC2 x 2   nCn x n
which extends to;
a  b n  nC0 a n  nC1a n1b nC2 a n2b 2   nCn1ab n1  nCnb n
e.g .2  3 x   4C0 2 4  4C1 23 3 x  4C2 2 2 3 x   4C3 23 x   4C4 3 x 
4 2 3 4

8. General Expansion of
Binomials
Ck is the coefficient of x k in 1  x 
n k
n
n
Ck   
k 
1  x n  nC0  nC1 x  nC2 x 2   nCn x n
which extends to;
a  b n  nC0 a n  nC1a n1b nC2 a n2b 2   nCn1ab n1  nCnb n
e.g .2  3 x   4C0 2 4  4C1 23 3 x  4C2 2 2 3 x   4C3 23 x   4C4 3 x 
4 2 3 4
 16  96 x  216 x 2  216 x 3  81x 4

9. Pascal’s Triangle Relationships

10. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1

11. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1

12. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 

13. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 
looking at coefficients of x k

14. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 
looking at coefficients of x k
LHS  nCk

15. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 
looking at coefficients of x k
LHS  nCk

16. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 
looking at coefficients of x k
LHS  nCk RHS  1 n1Ck 1   1 n1Ck 

17. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 
looking at coefficients of x k
LHS  nCk RHS  1 n1Ck 1   1 n1Ck 
 n1Ck 1  n1Ck

18. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 
looking at coefficients of x k
LHS  nCk RHS  1 n1Ck 1   1 n1Ck 
 n1Ck 1  n1Ck n Ck  n1Ck 1  n1Ck

19. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 
looking at coefficients of x k
LHS  nCk RHS  1 n1Ck 1   1 n1Ck 
 n1Ck 1  n1Ck n Ck  n1Ck 1  n1Ck
2 nCk  nCnk where 1  k  n  1
" Pascal' s triangle is symmetrical"

20. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 
looking at coefficients of x k
LHS  nCk RHS  1 n1Ck 1   1 n1Ck 
 n1Ck 1  n1Ck n Ck  n1Ck 1  n1Ck
2 nCk  nCnk where 1  k  n  1
" Pascal' s triangle is symmetrical"
3 nC0  nCn  1

21. Pascal’s Triangle Relationships
1 nCk  n1Ck 1  n1Ck where 1  k  n  1
1  x n  1  x 1  x n1
 1  x  n1C0  n1C1 x   n1Ck 1 x k 1  n1Ck x k   n1Cn1 x n1 
looking at coefficients of x k
LHS  nCk RHS  1 n1Ck 1   1 n1Ck 
 n1Ck 1  n1Ck n Ck  n1Ck 1  n1Ck
2 nCk  nCnk where 1  k  n  1
" Pascal' s triangle is symmetrical" Exercise 5B; 2ace, 5, 6ac,
10ac, 11, 14
3 nC0  nCn  1