Question
The co-efficient of $${x^3}{y^4}z$$ in the expansion of $${\left( {1 + x + y - z} \right)^9}$$ is
A.
$$2 \cdot {\,^9}{C_7} \cdot {\,^7}{C_4}$$
B.
$$- 2 \cdot {\,^9}{C_2} \cdot {\,^7}{C_3}$$
C.
$${\,^9}{C_7} \cdot {\,^7}{C_4}$$
D.
None of these
Answer :
$$- 2 \cdot {\,^9}{C_2} \cdot {\,^7}{C_3}$$
Solution :
$$\eqalign{
& {\left( {1 + x + y - z} \right)^9} = {\left\{ {\left( {1 - z} \right) + \left( {x + y} \right)} \right\}^9} \cr
& {\left( {1 + x + y - z} \right)^9} = {\,^9}{C_0}{\left( {1 - z} \right)^9} + {\,^9}{C_1}{\left( {1 - z} \right)^8}\left( {x + y} \right) + {\,^9}{C_2}{\left( {1 - z} \right)^7}{\left( {x + y} \right)^2} + ..... + {\,^9}{C_7}{\left( {1 - z} \right)^2} \cdot {\left( {x + y} \right)^7} + .....\,\,. \cr} $$
$${x^3}{y^4}$$ appears in $${\left( {x + y} \right)^7}$$ only.
∴ the required co-efficient $$ = {\,^9}{C_7}\left( { - 2} \right) \cdot {\,^7}{C_4}.$$