Question on finding the value of x

Question

If the coefficient of $x^2$ in the expansion of $(k+ \frac 1 3 x)^5$ is $30$. What is the value of the constant $k$?

Answer 1

Hint: The term in $t^2$ in the binomial expansion of $(s+t)^5$ is $\binom{5}{2}s^3t^2$. So the term in $x^2$ in the expansion of $\left(k+\frac{x}{3}\right)^2$ is $\binom{5}{2}k^3 \frac{x^2}{9}$. It follows that the coefficient of $x^2$ is $\frac{10k^3}{9}$.

Remark: The full expansion of $(s+t)^5$ is $$\binom{5}{0}s^5t^0+\binom{5}{1}s^4t+\binom{5}{2}s^3t^2+\binom{5}{3}s^2t^3+\binom{5}{4}st^4+\binom{5}{5}s^0t^5,$$ or more simply $s^5+5s^4t+10s^3t^2+10s^2t^3+5st^4+t^5$.

In your course, the binomial coefficient $\binom{5}{2}$ may be called $C^5_2$. or ${}^5C_2$, or $C(5,2)$. And there are other notations!