Explanation:

Let the total number of examinees be $T$.\nGiven that $65\%$ of the total examinees passed.\nTherefore, the percentage of examinees who failed is $100\% - 65\% = 35\%$. \nWe are also given that the number of failures is $420$.\nSo, we can set up the equation: $35\%$ of $T = 420$ $\frac{35}{100} \times T = 420$ \nTo find $T$, we rearrange the equation: $T = \frac{420 \times 100}{35}$ $T = \frac{420}{35} \times 100$ $T = 12 \times 100$ $T = 1200$. \nLet's evaluate the options:\nOption (1) $1400$: If $T=1400$, then $35\%$ of $1400 = 0.35 \times 1400 = 490$. This is not $420$.\nOption (2) $800$: If $T=800$, then $35\%$ of $800 = 0.35 \times 800 = 280$. This is not $420$.\nOption (3) $1200$: This matches our calculated total number of examinees. If $T=1200$, then $35\%$ of $1200 = 0.35 \times 1200 = 420$. This is correct.\nOption (4) $1000$: If $T=1000$, then $35\%$ of $1000 = 0.35 \times 1000 = 350$. This is not $420$. \nTherefore, the total number of examinees is $1200$. The correct option is (3) 1200.