Step 1: Understand the question.
A current of $1\,\text{A}$ flows for $16.1$ minutes. We must find how many electrons pass through.
Step 2: Find the charge.
Charge is current times time, $Q = I \times t$. First change minutes to seconds: $t = 16.1 \times 60 = 966\,\text{s}$. So \[ Q = 1 \times 966 = 966\ \text{C} \]
Step 3: Find moles of electrons.
One mole of electrons carries $96500\,\text{C}$ (Faraday's constant). So \[ n_{e} = \frac{966}{96500} \approx 0.01\ \text{mol} \]
Step 4: Bring in Avogadro's number.
Each mole has $6.022 \times 10^{23}$ electrons.
Step 5: Find the number of electrons.
\[ N = 0.01 \times 6.022 \times 10^{23} = 6.022 \times 10^{21} \]
Step 6: Pick the answer.
The number of electrons is $6.022 \times 10^{21}$, option (C).
\[ \boxed{6.022 \times 10^{21}\ \text{electrons}} \]