Question

Write the coefficient of a3b
in 8b3a3c​

Answer 1

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