Question

Find the coefficients of \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\] in the expansion of the following expression,
${\left[ {\left( {1 + {\text{x}}} \right)\left( {1 + {\text{y}}} \right)\left( {{\text{x + y}}} \right)} \right]^{\text{n}}}$.

Answer 1

Hint: In order to find the coefficients of \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\] from the given expansion, we use the formulae of binomial expansions of each of the individual terms from the given expansion, i.e., ${\left( {1 + {\text{x}}} \right)^{\text{n}}}$, ${\left( {1 + {\text{y}}} \right)^{\text{n}}}$ and ${\left( {{\text{x}} + {\text{y}}} \right)^{\text{n}}}$ respectively to figure out the coefficients.

Complete step-by-step solution:
Given Data,
Expansion - ${\left[ {\left( {1 + {\text{x}}} \right)\left( {1 + {\text{y}}} \right)\left( {{\text{x + y}}} \right)} \right]^{\text{n}}}$
Coefficients of \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\]
From the concept of binomial expansion, let us use the formula of binomial expansion of each term from the given equation,
We know the formula of the term ${\left( {1 + {\text{x}}} \right)^{\text{n}}}$ is given by
${\left( {1 + {\text{x}}} \right)^{\text{n}}} = {{\text{C}}_0} + {{\text{C}}_1}{\text{x + }}{{\text{C}}_2}{{\text{x}}^2} + ....... + {{\text{C}}_{\text{n}}}{{\text{x}}^{\text{n}}}$
We know the formula of the term ${\left( {1 + {\text{y}}} \right)^{\text{n}}}$ is given by
${\left( {1 + {\text{y}}} \right)^{\text{n}}} = {{\text{C}}_0}{{\text{y}}^{\text{n}}} + {{\text{C}}_1}{{\text{y}}^{{\text{n - 1}}}}{\text{ + }}{{\text{C}}_2}{{\text{y}}^{{\text{n - }}2}} + ....... + {{\text{C}}_{\text{n}}}$
We know the formula of the term ${\left( {{\text{x}} + {\text{y}}} \right)^{\text{n}}}$ is given by
${\left( {{\text{x}} + {\text{y}}} \right)^{\text{n}}} = {{\text{C}}_0}{{\text{x}}^{\text{n}}} + {{\text{C}}_1}{{\text{x}}^{{\text{n - 1}}}}{\text{y + }}{{\text{C}}_2}{{\text{x}}^{{\text{n - }}2}}{{\text{y}}^2} + ....... + {{\text{C}}_{\text{n}}}{{\text{y}}^{\text{n}}}$
If we multiply all these binomial expansions of terms, each vertical row will be the coefficient of the term \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\], i.e.
The first terms of all the equations, gives a variable value of \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\] with a coefficient of\[{{\text{C}}_{\text{0}}}^{\text{3}}\].
The coefficient of the first term of the equations is ${{\text{C}}_{\text{0}}} \times {{\text{C}}_{\text{0}}}{{\text{y}}^{\text{n}}} \times {{\text{C}}_{\text{0}}}{{\text{x}}^{\text{n}}} = {{\text{C}}_{\text{0}}}^{\text{3}}{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}$is${{\text{C}}_0}^{\text{3}}$.
When we multiply the expansions of ${\left( {1 + {\text{x}}} \right)^{\text{n}}}$,${\left( {1 + {\text{y}}} \right)^{\text{n}}}$ and ${\left( {{\text{x}} + {\text{y}}} \right)^{\text{n}}}$, the first, second and so on nth terms of the equations will contain the same variable term which is \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\] and a coefficient ${{\text{C}}_0}^{\text{3}}$,${{\text{C}}_1}^{\text{3}}$…….${{\text{C}}_{\text{n}}}^{\text{3}}$ for the first, second and so on the nth term of the product equation of all three expansions.
Similarly the coefficient of \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\] for the last term of the product of binomial expansions of each term will also be ${{\text{C}}_{\text{n}}}^{\text{3}}$.
Therefore the coefficients of \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\] in the expansion ${\left[ {\left( {1 + {\text{x}}} \right)\left( {1 + {\text{y}}} \right)\left( {{\text{x + y}}} \right)} \right]^{\text{n}}}$ is given as:
${{\text{C}}_0}^{\text{3}} + {{\text{C}}_1}^{\text{3}} + {{\text{C}}_2}^{\text{3}} + ...... + {{\text{C}}_{\text{n}}}^{\text{3}}$.

Note: In order to solve this type of problem the key is to know the concept of binomial expansion and the respective formulae of binomial expansion to the power of n for each of the term in the given expansion. The key step in solving this problem is identifying product of each of the respective term in each binomial expansion results in the forming term \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\]. So the sum of all the individual coefficients gives us the coefficient of \[{{\text{x}}^{\text{n}}}{{\text{y}}^{\text{n}}}\] for the given expansion.