Find the ratio of the coefficients of

Question:

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

Solution:

Coefficient of $x^{p}$ in the expansion of $(1+x)^{p+q}$ is $^{p+q} C_{p}$.

Coefficient of $x^{q}$ in the expansion of $(1+x)^{p+q}$ is $^{p+q} C_{q}$.

Now,

$\frac{p+q_{p}}{p+q_{q}}=\frac{\frac{(p+q) !}{p l q !}}{\frac{(p+q) !}{q ! p !}}=1$

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

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