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
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
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