Question

The coefficient of ${\mathrm{p}}^{\mathrm{n}}{\mathrm{q}}^{\mathrm{n}}$ in the expansion of ${\left[\left(1+\mathrm{p}\right)\left(1+\mathrm{q}\right)\left(\mathrm{p}+\mathrm{q}\right)\right]}^{\mathrm{n}}$ is

• A. $\sum _{k=0}^n{\left[\mathrm{C}\left(n,k\right)\right]}^2$
• B. $\sum _{k=0}^n{\left[\mathrm{C}\left(n,k+2\right)\right]}^2$
• C. $\sum _{k=0}^n{\left[\mathrm{C}\left(n,k+3\right)\right]}^2$
• D. $\sum _{k=0}^n{\left[\mathrm{C}\left(n,k\right)\right]}^3$

Answer 1

The correct option is D

$\sum _{k=0}^n{\left[\mathrm{C}\left(n,k\right)\right]}^3$

Explanation for the correct option:

Given, the expression is ${\left[\left(1+\mathrm{p}\right)\left(1+\mathrm{q}\right)\left(\mathrm{p}+\mathrm{q}\right)\right]}^{\mathrm{n}}$.

Here, we use the principle of mathematical Induction,

Therefore, by using the principle of mathematical induction, we get

${\left[\left(1+\mathrm{p}\right)\left(1+\mathrm{q}\right)\left(\mathrm{p}+\mathrm{q}\right)\right]}^{\mathrm{n}}$,

Case 1: For $\mathrm{n}=1$

$\Rightarrow {\left[\left(1+\mathrm{p}\right)\left(1+\mathrm{q}\right)\left(\mathrm{p}+\mathrm{q}\right)\right]}^1$$=\left(1+\mathrm{p}+\mathrm{q}+\mathrm{pq}\right)\left(\mathrm{p}+\mathrm{q}\right)$

$=\mathrm{p}+\mathrm{q}+2\mathrm{pq}+{\mathrm{p}}^2+{\mathrm{q}}^2+{\mathrm{p}}^2\mathrm{q}+{\mathrm{pq}}^2$

Therefore, the coefficient of $\mathrm{pq}=2$

$={\left[\mathrm{C}\left(2,1\right)\right]}^3+{\left[\mathrm{C}\left(1,1\right)\right]}^3$

Case 2: for $\mathrm{n}=2$,

$\Rightarrow {\left[\left(1+\mathrm{p}\right)\left(1+\mathrm{q}\right)\left(\mathrm{p}+\mathrm{q}\right)\right]}^2={\left(1+\mathrm{p}\right)}^2{\left(1+\mathrm{q}\right)}^2{\left(\mathrm{p}+\mathrm{q}\right)}^2$

$=\left(1+{\mathrm{q}}^2+2\mathrm{q}\right)\left(1+{\mathrm{p}}^2+2\mathrm{p}\right)\left({\mathrm{p}}^2+{\mathrm{q}}^2+2\mathrm{pq}\right)$

$=\left(1+{\mathrm{q}}^2+2\mathrm{q}+{\mathrm{p}}^2{\mathrm{q}}^2+2{\mathrm{p}}^2\mathrm{q}+2\mathrm{p}+2{\mathrm{pq}}^2+4\mathrm{pq}\right)\left({\mathrm{p}}^2+{\mathrm{q}}^2+2\mathrm{pq}\right)$

Therefore, the coefficient of ${\mathrm{p}}^2{\mathrm{q}}^2=10$

$\Rightarrow {\left[\mathrm{C}\left(2,0\right)\right]}^3+{\left[\mathrm{C}\left(2,2\right)\right]}^3+{\left[\mathrm{C}\left(2,1\right)\right]}^3=\sum _{k=0}^2{\left[\mathrm{C}\left(2,\mathrm{k}\right)\right]}^3$

Similarly, the coefficient of ${\mathrm{p}}^{\mathrm{n}}{\mathrm{q}}^{\mathrm{n}}$ is $\sum _{k=0}^n{\left[\mathrm{C}\left(n,k\right)\right]}^3$

Hence, option(D) is the correct answer.