Question

Find the ratio of the coefficients of x^p and x^q in the expansion of ${\left(1+x\right)}^{p+q}$.

Answer 1

Coefficient of x^p in the expansion of ${\left(1+x\right)}^{p+q}$ is ${}^{p+q}{C}_p$.

Coefficient of x^q in the expansion of ${\left(1+x\right)}^{p+q}$ is ${}^{p+q}{C}_q$.

Now,
$\frac{{}^{p+q}{C}_p}{{}^{p+q}{C}_q}=\frac{{\displaystyle \frac{\left(p+q\right)!}{p!q!}}}{{\displaystyle \frac{\left(p+q\right)!}{q!p!}}}=1$

Hence, the ratio of the coefficients of x^p and x^q in the expansion of ${\left(1+x\right)}^{p+q}$ is 1 : 1.