Write the coefficient of a3b
in 8b3a3c
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Let us consider coefficient of x
p
y
q
r
n−p−q
in the expansion of (x+y+z)
n
We have p≥0,n≥p+q.
Now we can get p in
n
C
p
ways and from remaining (n−p) ways we can get q in
n−p
C
q
Hence x
p
y
q
z
r
can occur in
p!q!(n−p−q)!
n!
ways.
Consider (ab+bc+ca)
6
=(abc)
6
(
abc
ab+bc+ca
)
6
=(abc)
6
(
a
1
+
b
1
+
c
1
)
6
And we are seeking coefficient of a
3
b
4
c
5
=(abc)
6
(
a
1
)
3
(
b
1
)
2
(
c
1
)
If x=
a
1
,y=
b
1
and z=
c
1
Coefficent of x
3
y
2
z in (x+y+z)
6
=
3!2!1!
6!
=
3×2×1×2×1
6×5×4×3×2×1
=60
p
y
q
r
n−p−q
in the expansion of (x+y+z)
n
We have p≥0,n≥p+q.
Now we can get p in
n
C
p
ways and from remaining (n−p) ways we can get q in
n−p
C
q
Hence x
p
y
q
z
r
can occur in
p!q!(n−p−q)!
n!
ways.
Consider (ab+bc+ca)
6
=(abc)
6
(
abc
ab+bc+ca
)
6
=(abc)
6
(
a
1
+
b
1
+
c
1
)
6
And we are seeking coefficient of a
3
b
4
c
5
=(abc)
6
(
a
1
)
3
(
b
1
)
2
(
c
1
)
If x=
a
1
,y=
b
1
and z=
c
1
Coefficent of x
3
y
2
z in (x+y+z)
6
=
3!2!1!
6!
=
3×2×1×2×1
6×5×4×3×2×1
=60
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