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.
\[\begin{gathered}
  {\left( {1 - ax} \right)^{ - 1}} = 1 + ax + \dfrac{{\left( { - 1} \right)\left( { - 2} \right)}}{{2!}}{\left( { - ax} \right)^2} + \ldots \\
  {\left( {1 - bx} \right)^{ - 1}} = 1 + bx + \dfrac{{\left( { - 1} \right)\left( { - 2} \right)}}{{2!}}{\left( { - bx} \right)^2} + \ldots \\
  {\left( {1 - cx} \right)^{ - 1}} = 1 + cx + \dfrac{{\left( { - 1} \right)\left( { - 2} \right)}}{{2!}}{\left( { - cx} \right)^2} + \ldots \\
\end{gathered}\]

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.