Given:
[tex](3x-2)^9[/tex]
To find:
The coefficient of [tex]x^6[/tex] in the expansion of [tex](3x-2)^9[/tex].
Solution:
We know that, in the expansion of [tex](x+y)^n[/tex],
[tex]T_{r+1}=^nC_rx^{n-r}y^r[/tex] ...(i)
[tex](3x-2)^9[/tex] has only one x and in [tex]x^6[/tex], the power of x is 6. It is possible if
[tex]n-r=6[/tex]
[tex]9-r=6[/tex]
[tex]9-6=r[/tex]
[tex]3=r[/tex]
Putting n=9, r=3, x=3x and y=-2 in (i), we get
[tex]T_{3+1}=^9C_3(3x)^{9-3}(-2)^3[/tex]
[tex]T_{4}=\dfrac{9!}{3!(9-3)!}\times (3x)^{6}(-8)[/tex]
[tex]T_{4}=\dfrac{9\times 8\times 7\times 6!}{3\times 2\times 1\times (9-3)!}\times (729x^{6})(-8)[/tex]
[tex]T_{4}=84\times (729)\times (-8)\times x^{6}[/tex]
[tex]T_{4}=-489888x^6[/tex]
Therefore, the coefficient of [tex]x^6[/tex] in the expansion of [tex](3x-2)^9[/tex] is -489888.
