1. For any gas at the same temperature, the root mean square speed is $v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$ where $M$ is the molar mass.

2. Since helium and oxygen are in the same sample, they are at the same temperature. Therefore, $\frac{v_{\text{rms, He}}}{v_{\text{rms, O}_2}}=\sqrt{\frac{M_{\text{O}_2}}{M_{\text{He}}}}$

3. Molar masses:

• Helium: $M_{\text{He}}=4$
• Oxygen gas: $M_{\text{O}_2}=32$

So, $\frac{v_{\text{rms, He}}}{v_{\text{rms, O}_2}}=\sqrt{\frac{32}{4}}=\sqrt{8}=2\sqrt{2}$

4. Hence the required ratio is $2\sqrt{2}:1$

5. Checking options:

• A: $\frac{1}{2\sqrt{2}}$ ❌
• B: $\frac{1}{4}$ ❌
• C: $\frac{2\sqrt{2}}{1}$ ✅
• D: $\frac{1}{32}$ ❌

Therefore, the correct option is C.