Step 1: Understand half-life.
Half-life is the time in which a radioactive sample falls to half of its amount. After each half-life period passes, whatever is left becomes half again. This repeated halving is the key idea here.
Step 2: Write the decay formula.
After $n$ half-lives the remaining amount is $N=N_0\left(\tfrac12\right)^n$, where $N_0$ is the starting amount and $n=\frac{t}{T_{1/2}}$ is the number of half-lives that fit in the total time.
Step 3: Count the half-lives.
The half-life is $150$ days and the total time is $600$ days. \[ n=\frac{600}{150}=4 \] So four halvings happen.
Step 4: Start the halving.
We begin with $N_0=1\,\text{g}$. After the first half-life we have $\frac12$ g, after the second $\frac14$ g, after the third $\frac18$ g.
Step 5: Finish the last halving.
After the fourth half-life, halve $\frac18$ once more to get $\frac{1}{16}$ g. The same comes from the formula. \[ N=1\times\left(\tfrac12\right)^4=\frac{1}{16}\,\text{g} \]
Step 6: State the result.
After $600$ days, only $\frac{1}{16}\,\text{g}$ of the element remains, which is option 4. \[ \boxed{\tfrac{1}{16}\ \text{g}} \]