How many terms are rational in the binomial expansion of $(x^{1/2}+y^{1/3})^{100}$?

Question

I replaced $100$ with $n$, and for there to be a rational term, $n$ must be even or a multiple of $3$. There are 50 even numbers between 1 and 100, and 33 multiples of 3. You must subtract the multiples of 6 because they are double counted.
$50+33-16=67$ rational terms. Am I correct?

Answer 1

Hint:

As $\;\bigl(x^{1/2}+y^{1/3}\bigr)^{100}=\displaystyle\sum_{k=0}^{100}\dbinom{100}{k}x^{k/2}y^{(100-k)/3}$, you have to solve the system $$\begin{cases}k\equiv 0\mod 2\\k\equiv 100\equiv 1\mod 3\end{cases}\quad\text{in natural numbers},$$ i.e. in the arithmetic progression $1+3m$, you have to find the even terms less than $100$.