Consider the cyclic redundancy check (CRC) based error detecting scheme having the generator polynomial $X^3 + X + 1$. Suppose the message $m_4 m_3 m_2 m_1 m_0 = 11000$ is to be transmitted. Check bits $c_2 c_1 c_0$ are appended at the end of the message by the transmitter using the above CRC scheme. The transmitted bit string is denoted by $m_4 m_3 m_2 m_1 m_0 c_2 c_1 c_0$. The value of the checkbit sequence $c_2 c_1 c_0$ is

To determine the check bit sequence $c_2 c_1 c_0$:

1. Find the Generator Bit String: The generator polynomial is $G(X) = X^3 + X + 1 = 1 \cdot X^3 + 0 \cdot X^2 + 1 \cdot X^1 + 1 \cdot X^0$. Thus, the generator bit pattern is $G = 1011$.

2. Append Zeros to the Message: The degree of $G(X)$ is $r = 3$. We append $r = 3$ zeros to the message string $M = 11000$:

   $$M^{\prime} = 11000000$$

3. Perform Modulo-2 Division of $M^{\prime}$ by $G$: $$\begin{array}{r} 11000000 \\ \oplus 1011\phantom{0000} \\ \hline 01110\phantom{000} \\ \oplus \phantom{0}1011\phantom{000} \\ \hline 01010\phantom{00} \\ \oplus \phantom{0}1011\phantom{00} \\ \hline 0001000 \\ \end{array}$$ Since the remaining leading bits after the last subtraction cannot be divided further by $1011$, bringing down the remaining two zeros gives a $3$-bit remainder of $100$.

Thus, the CRC checkbit sequence $c_2 c_1 c_0 = 100$, which corresponds to Option (C).