Question

The coefficients of $x^p$ and$x^q$ (p and q are positive integers) in the expansion of $(1+x{)}^{p+q}$ are

• A. Equal
• B. Equal with opposite signs
• C. Reciprocals to each other
• D. None of these

Answer 1

The correct option is A Equal

$(1+x{)}^{p+q}$

General term$=C_r^{p+q}$

$\Rightarrow x^{p+q-r}=x^p$

$\Rightarrow r=q$

Co-efficient of $x^p=C_q^{p+q}=\frac{(p+q)!}{p!q!}$

$x^{p+q-r}=x^q$

$\Rightarrow r=p$

Co-efficient of $x^q=C_q^{p+q}=\frac{(p+q!)}{p!q!}$

$\therefore$ Both co-efficients are equal.