Compute the number of permutations $(a,b,c,x,y,z)$ of $(1,2,3,4,5,6)$ which satisfy the five inequalities
  \[ a < b < c, \quad x < y < z, \quad a < x, \quad b < y, \quad\text{and}\quad c < z. \]

[i]Proposed by Evan Chen[/i]
To solve the problem, we need to count the number of permutations \((a, b, c, x, y, z)\) of \((1, 2, 3, 4, 5, 6)\) that satisfy the following inequalities:
\[ a < b < c, \quad x < y < z, \quad a < x, \quad b < y, \quad c < z. \]

1. **Identify the constraints:**
   - \(a < b < c\)
   - \(x < y < z\)
   - \(a < x\)
   - \(b < y\)
   - \(c < z\)

2. **Determine the possible values for \(a\) and \(z\):**
   - Since \(a < x\) and \(c < z\), the smallest value \(a\) can take is 1, and the largest value \(z\) can take is 6. Therefore, \(a = 1\) and \(z = 6\).

3. **List the remaining values:**
   - The remaining values are \(\{2, 3, 4, 5\}\).

4. **Assign values to \(b\) and \(c\):**
   - We need to choose 2 values from \(\{2, 3, 4, 5\}\) for \(b\) and \(c\) such that \(b < c\). The number of ways to choose 2 values from 4 is given by the binomial coefficient \(\binom{4}{2}\), and for each pair, there is only one way to order them such that \(b < c\).

5. **Assign values to \(x\) and \(y\):**
   - The remaining 2 values will be assigned to \(x\) and \(y\) such that \(x < y\). Since \(a < x\) and \(b < y\) are already satisfied by the choice of \(a\) and \(b\), we only need to ensure \(x < y\).

6. **Calculate the number of valid permutations:**
   - The number of ways to choose 2 values from 4 is \(\binom{4}{2} = 6\).
   - For each choice of \(b\) and \(c\), there is exactly one way to assign the remaining values to \(x\) and \(y\) such that \(x < y\).

Therefore, the total number of valid permutations is:
\[ \boxed{6} \]