To determine if the system of equations:

$$\begin{aligned} 2x + 7y + \lambda z = 3 \\ 3x + 2y + 5z = 4 \\ x + \mu y + 32z = -1 \end{aligned}$$

has infinitely many solutions, we must use Cramer's rule.

The determinants are calculated as follows:

$$\begin{aligned} \Delta &= -2\lambda + 3\lambda\mu - 10\mu - 509 \\ \Delta_1 &= 2\lambda + 3\lambda\mu - 15\mu - 739 \\ \Delta_2 &= -7\lambda - 7 \\ \Delta_3 &= \mu + 39 \end{aligned}$$

To have infinitely many solutions, the determinants must satisfy:

$$\begin{aligned} \Delta = \Delta_1 = \Delta_2 = \Delta_3 = 0 \end{aligned}$$

Solving these equations, we find:

$$\lambda = -1, \mu = -39$$

Thus, the value of $ \lambda - \mu $ is:

$$ \lambda - \mu = 38 $$