Question

Find the coeifficient of
(i) $x^6{y}^{3}in(x+y{)}^9$

Answer 1

$x^6{y}^3$ in $(x+y{)}^9$

We know general term of expansion $(a+b{)}^n$ is

$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$

For $(x+y{)}^9$

Putting $n=9,a=x,b=y$

$T_{r+1}={}^9{C}_r\left(x{)}^{9-r}\right(y{)}^r$ ___(i)

we need to find coefficient of $x^6{y}^3$

Comparing, $y^r=y^3$

$r=3$

Putting $r=3$ in (i)

$T_{3+1}{=}^9{C}_3\left(x{)}^{9-3}\right(y{)}^3$

$=\frac{9!}{3!6!}\left(x{)}^6\right(y{)}^3$

$=\frac{9\times 7\times 6!}{3\times 2\times 6!}\left(x{)}^6\right(y{)}^3$

$=\frac{3\times 7}{2}\left(x{)}^6\right(y{)}^3$

$=\frac{21}{2}\left(x{)}^6\right(y{)}^3$

Hence, coefficient of $x^6{y}^3$ is $\frac{21}{2}$