Find the coefficient x to the power 4 * y cube & number of terms in the expansion of (3x – 10y)11

Expansion of $(3x – 10y)^{11}$ $= C(11,0)\times (3x)^{11} \times (-10y)^0 + C(11,1) \times (3x)^{10} \times (-10y)^1+ C(11,2) \times (3x)^{9} \times (-10y)^2 + ... + C(11,10) \times (3x)^{1} \times (-10y)^{10} + C(11,11) \times (3x)^0 \times (-10y)^{11}$

$= 3^{11} x^{11} -(11 \times 3^{10} \times 10^1) x^{10} y^1 + (55 \times 3^9 \times 10^2) x^9 y^2 + ... + (11 \times 3^{1} \times 10^{10}) x^{1} y^{10} - 10^{11} y^{11}$

General term is $T_{r+1} = C(11,r) (3x)^{11-r} (-10y)^{r}$ .

Hence, number of terms are 12.

Term containing $y^3$ appears when $r = 3$ i.e. fourth term.

So, $T_4 = C(11,3) (3x)^8 (-10y)^3 = (165000 \times 3^8) x^8 y^3$ .

This term is not of $x^4 y^3$ form, hence coefficient of $x^4 y^3$ is zero.