Find the coefficient of the x^9in the expansion of (3x* - 1/x)^6

Calculate the binomial coefficient and appropriate powers of
2
and
3
to find that the required coefficient is:
1088640

Explanation:
(
2
x
+
3
)
10
=
10
∑
k
=
0

(
10
k
)
2
10
−
k
3
k
x
10
−
k

The term in
x
6
is the one for
k
=
4
, so has coefficient:
(
10
4
)
2
6
⋅
3
4

=
10
!
4
!
6
!
⋅
64
⋅
81

=
10
×
9
×
8
×
7
4
×
3
×
2
×
1
⋅
64
⋅
81

=
210
⋅
64
⋅
81
=
1088640

Instead of calculating
(
10
4
)
, you can pick it out from the appropriate row of Pascal's triangle...
enter image source here

Just divide the x axis (the top) by 2 or multiply the y axis (the bottom) by 2