Question

The coefficient of ${a^{10}}{b^7}{c^3}$ in the expansion of ${\left( {bc + ca + ab} \right)^{10}}$is
A. $30$
B. $60$
C. $120$
D. None of these

Answer 1

As we know a coefficient is the numerical factor of a term containing constant and variables or we can say coefficient is an integer that is written along with a variable or it is multiplied by the variable Consider the three variables to expand the equation ${\left( {bc + ca + ab} \right)^{10}}$ as,
${\left( {abc} \right)^{10}}{\left( {\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}} \right)^{10}}$.

Complete step by step answer:
To find the coefficient of the given term, let us consider the general term to get the value of the given expansion.
The general term in the expansion of ${\left( {bc + ca + ab} \right)^{10}}$ is
${\left( {abc} \right)^{10}}{\left( {\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}} \right)^{10}}$
Simplifying the terms, we get
${\left( {\dfrac{1}{0} + \left( {\dfrac{1}{b} + \dfrac{1}{c}} \right)} \right)^{10}}$
In which,
${T_{r + 1}} = {}^{10}{C_r}{\left( {\dfrac{1}{a}} \right)^r}{\left( {\dfrac{1}{b} + \dfrac{1}{c}} \right)^{10 - r}}$
As from the given expression ${a^{10}}$ is common, so let the value of a be the same and no need to calculate again the value of a.
Equating r=0, implying we get
${T_1}{}^{10}{C_0}{\left( {\dfrac{1}{b} + \dfrac{1}{c}} \right)^{10}}$
After simplifying the terms, we get as
${T_1}{}^{10}{C_0}{\left( {\dfrac{1}{b} + \dfrac{1}{c}} \right)^{10}} = {\left( {\dfrac{1}{b} + \dfrac{1}{c}} \right)^{10}}$
Hence, the general term of ${T_1}{}^{10}{C_0}$ is
${T_{j + 1}} = {}^{10}{C_j}{\left( {\dfrac{1}{b}} \right)^j}{\left( {\dfrac{1}{c}} \right)^{10 - j}}$
Next for ${b^7}$,
$\dfrac{1}{{{b^3}}} \Rightarrow j = 3$
Thus, the coefficient of ${a^{10}}{b^7}{c^3}$ in the expansion of ${\left( {bc + ca + ab} \right)^{10}}$ is
${}^{10}{C_3} = \dfrac{{10 \times 9 \times 8}}{{1 \times 3 \times 2}}$
Thus, the value of the combination term is
${}^{10}{C_3} = 120$

Therefore, option $C$ is the right answer.

Additional information:
Here are some of the definitions we must know
A coefficient is the numerical factor of a term containing constant and variables or we can say coefficient is an integer that is written along with a variable or it is multiplied by the variable and the variables which do not carry any number along with them, have a coefficient 1. For example, the term x has coefficient 1.

Note:
The key point to find the coefficient of equation given, combine the terms and find the combination of the terms and as we know that in combination there are no repetitions of objects allowed and order is not important to find a combination. A term can be a number, a variable, product of two or more variables or product of a number and a variable. An algebraic expression is formed by a single term or by a group of terms.