Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

$ \textbf{(A)}\  21 \qquad \textbf{(B)}\  60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\  180\qquad \textbf{(E)}\ 231$
1. Let the two different prime numbers be \( x \) and \( y \). We are given that these primes are between 4 and 18. The prime numbers in this range are: 5, 7, 11, 13, and 17.
2. We need to find the value of \( xy - (x + y) \). This can be rewritten as:
   \[
   xy - x - y = (x-1)(y-1) - 1
   \]
3. We need to check which of the given options (21, 60, 119, 180, 231) can be expressed in the form \((x-1)(y-1) - 1\).

4. Since \( x \) and \( y \) are both prime and greater than 2, they are odd. Therefore, \( x-1 \) and \( y-1 \) are even numbers. This means \((x-1)(y-1)\) is a product of two even numbers, which is even. Subtracting 1 from an even number results in an odd number. Thus, the expression \((x-1)(y-1) - 1\) must be odd.

5. From the given options, the odd numbers are 21, 119, and 231. We will check these values:
   - For 21: 
     \[
     (x-1)(y-1) - 1 = 21 \implies (x-1)(y-1) = 22
     \]
     The factor pairs of 22 are (2, 11) and (11, 2). Neither of these pairs correspond to \((x-1)\) and \((y-1)\) where \( x \) and \( y \) are primes between 4 and 18.
   - For 119:
     \[
     (x-1)(y-1) - 1 = 119 \implies (x-1)(y-1) = 120
     \]
     The factor pairs of 120 are (2, 60), (4, 30), (6, 20), (8, 15), (10, 12), and their reverses. Checking these pairs:
     - \( x-1 = 10 \) and \( y-1 = 12 \) gives \( x = 11 \) and \( y = 13 \), both of which are primes.
     - Therefore, \( x = 11 \) and \( y = 13 \) works, and we get:
       \[
       (11-1)(13-1) - 1 = 10 \cdot 12 - 1 = 120 - 1 = 119
       \]
   - For 231:
     \[
     (x-1)(y-1) - 1 = 231 \implies (x-1)(y-1) = 232
     \]
     The factor pairs of 232 are (2, 116), (4, 58), (8, 29), and their reverses. None of these pairs correspond to \((x-1)\) and \((y-1)\) where \( x \) and \( y \) are primes between 4 and 18.

6. Therefore, the only valid solution is when the expression equals 119.

The final answer is \(\boxed{119}\).