Question

The coefficient of x in the expansion of ${\left( {1 - ax} \right)^{ - 1}}{\left( {1 - bx} \right)^{ - 1}}{\left( {1 - cx} \right)^{ - 1}}$ is?
A $a + b + c$
B $a - b – c$
C $- a + b + c$
D $a - b + c$

Answer 1

Hint: In this problem, first we need to find the binomial expansion of individual expressions. Now, collect the coefficient of $x$ from the binomial expansion. The binomial expansion is the algebraic expansion of powers of a binomial.

Complete step-by-step answer:
The binomial expansion of the expressions ${\left( {1 - ax} \right)^{ - 1}}$, ${\left( {1 - bx} \right)^{ - 1}}$ and ${\left( {1 - cx} \right)^{ - 1}}$ are shown below.

{\left( {1 - ax} \right)^{ - 1}} = 1 + ax + \dfrac{{\left( { - 1} \right)\left( { - 2} \right)}}{{2!}}{\left( { - ax} \right)^2} + \ldots \\\ {\left( {1 - bx} \right)^{ - 1}} = 1 + bx + \dfrac{{\left( { - 1} \right)\left( { - 2} \right)}}{{2!}}{\left( { - bx} \right)^2} + \ldots \\\ {\left( {1 - cx} \right)^{ - 1}} = 1 + cx + \dfrac{{\left( { - 1} \right)\left( { - 2} \right)}}{{2!}}{\left( { - cx} \right)^2} + \ldots \\\ \end{gathered}$$ Now, the coefficient of $$x$$ in expression $${\left( {1 - ax} \right)^{ - 1}}{\left( {1 - bx} \right)^{ - 1}}{\left( {1 - cx} \right)^{ - 1}}$$ can be calculated as shown below. $$\begin{gathered} \,\,\,\,\,\,{\left( {1 - ax} \right)^{ - 1}}{\left( {1 - bx} \right)^{ - 1}}{\left( {1 - cx} \right)^{ - 1}} \\\ \Rightarrow \left( {1 + ax + \dfrac{{\left( { - 1} \right)\left( { - 2} \right)}}{{2!}}{{\left( { - ax} \right)}^2} + \ldots } \right)\left( {1 + bx + \dfrac{{\left( { - 1} \right)\left( { - 2} \right)}}{{2!}}{{\left( { - bx} \right)}^2} + \ldots } \right)\left( {1 + cx + \dfrac{{\left( { - 1} \right)\left( { - 2} \right)}}{{2!}}{{\left( { - cx} \right)}^2} + \ldots } \right) \\\ \end{gathered}$$ Collect the coefficient of $$x$$ in the above expression. $$\begin{gathered} \,\,\,\,{\text{Coefficient of }}x = \left( a \right)\left( 1 \right)\left( 1 \right) + \left( 1 \right)\left( b \right)\left( 1 \right) + \left( 1 \right)\left( 1 \right)\left( c \right) \\\ \Rightarrow {\text{Coefficient of }}x = a + b + c \\\ \end{gathered}$$ Thus, the coefficient of $$x$$ is $$a + b + c$$, hence, the option (A) is the correct answer. Note: Binomial expansion is used to find the algebraic expansion of powers of a binomial. While collecting the coefficient of $$x$$, only collect those terms which are multiple of $$x$$ only.