Consider all words constituted by eight letters from $\{C ,H,M, O\}$. We arrange the words in an alphabet sequence.
Precisely, the first word is $CCCCCCCC$, the second one is $CCCCCCCH$, the third is $CCCCCCCM$, the fourth one is $CCCCCCCO,  ...,$ and the last word is $OOOOOOOO$.
a) Determine the $2017$th word of the sequence?
b) What is the position of the word $HOMCHOMC$ in the sequence?
To solve this problem, we need to understand how to convert between base-4 and base-10, as the letters $\{C, H, M, O\}$ can be mapped to the digits $\{0, 1, 2, 3\}$ respectively. 

### Part (a)
1. **Convert 2017 from base-10 to base-4:**
   - We need to find the base-4 representation of 2017.
   - The base-4 representation is found by repeatedly dividing the number by 4 and recording the remainders.

\[
\begin{align*}
2017 \div 4 &= 504 \quad \text{remainder} \ 1 \\
504 \div 4 &= 126 \quad \text{remainder} \ 0 \\
126 \div 4 &= 31 \quad \text{remainder} \ 2 \\
31 \div 4 &= 7 \quad \text{remainder} \ 3 \\
7 \div 4 &= 1 \quad \text{remainder} \ 3 \\
1 \div 4 &= 0 \quad \text{remainder} \ 1 \\
\end{align*}
\]

   - Reading the remainders from bottom to top, we get \(2017_{10} = 130231_4\).

2. **Map the base-4 digits to letters:**
   - Using the mapping \(0 \rightarrow C\), \(1 \rightarrow H\), \(2 \rightarrow M\), \(3 \rightarrow O\), we convert \(130231_4\) to letters:
   - \(130231_4 \rightarrow HOOHMC\).

3. **Adjust for the sequence starting at 0:**
   - Since the sequence starts at 0, the 2017th word corresponds to the 2018th item in the sequence.
   - Therefore, the 2017th word is \(HOOHMC\).

### Part (b)
1. **Convert the word \(HOMCHOMC\) to base-4:**
   - Using the mapping \(C \rightarrow 0\), \(H \rightarrow 1\), \(M \rightarrow 2\), \(O \rightarrow 3\), we convert \(HOMCHOMC\) to digits:
   - \(HOMCHOMC \rightarrow 13021302_4\).

2. **Convert the base-4 number to base-10:**
   - We need to convert \(13021302_4\) to base-10.
   - The base-10 value is calculated as follows:

\[
\begin{align*}
1 \cdot 4^7 + 3 \cdot 4^6 + 0 \cdot 4^5 + 2 \cdot 4^4 + 1 \cdot 4^3 + 3 \cdot 4^2 + 0 \cdot 4^1 + 2 \cdot 4^0 &= \\
1 \cdot 16384 + 3 \cdot 4096 + 0 \cdot 1024 + 2 \cdot 256 + 1 \cdot 64 + 3 \cdot 16 + 0 \cdot 4 + 2 \cdot 1 &= \\
16384 + 12288 + 0 + 512 + 64 + 48 + 0 + 2 &= \\
29298.
\end{align*}
\]

3. **Adjust for the sequence starting at 0:**
   - Since the sequence starts at 0, the position of the word \(HOMCHOMC\) is 29299.

The final answer is \( \boxed{ HOOHMC } \) for part (a) and 29299 for part (b).