Question

The coefficient of $x^p$ & $x^q$ $(p,qϵN)$ in the expansion of ${(1+x)}^{p+q}$ is

• A. Reciprocal of each other
• B. Equal with opposite sign
• C. Equal
• D. None of these

Answer 1

The correct option is C Equal

$(1+x{)}^{p+q}$
$=(1+x{)}^p(1+x{)}^q$
$=(1+px+\frac{p(p-1)}{2!}{x}^2...+x^p)(1+qx+\frac{q(q-1)}{2!}{x}^2...+x^q)$
Therefore the coefficient of $x^p$ will be 1.
Similarly the coefficient of $x^q$ will be 1.
Hence the coefficients of $x^p$ and $x^q$ are equal.