Question

The coefficient of  x^3 in the expansion of  m^3(1-x/m)^3 is:​

Answer 1

The $(r+1)^{th}$ term in the expansion of $m^3\left(1-\dfrac{x}{m}\right)^3=(m-x)^3$ is,

$\longrightarrow T_{r+1}=\,^3C_r\ m^{3-r}\ (-x)^r$

$\longrightarrow T_{r+1}=(-1)^r\cdot\,^3C_r\ m^{3-r}\ x^r$

The coefficient in the $(r+1)^{th}$ term,

$\longrightarrow C_{r+1}=(-1)^r\cdot\,^3C_r\ m^{3-r}$

To find coefficient of $x^3,$ we need to equate the exponent of $x$ to 3, i.e.,

$\longrightarrow r=3$

So the 4th term contains $x^3,$ and the coefficient of $x^3$ is,

$\longrightarrow C_4=(-1)^3\cdot\,^3C_3\ m^{3-3}$

$\longrightarrow\underline{\underline{C_4=-1}}$

Hence -1 is the answer.