Value of Parameters for Infinitely Many Solutions in System of Equations

To find the values of $\lambda$ and $\mu$ for which the given system of linear equations has infinitely many solutions, we observe the linear combinations of the equations.

The given system is: $x + y + z = 6 \quad \text{--- (1)}$ $x + 2y + 5z = 10 \quad \text{--- (2)}$ $2x + 3y + \lambda z = \mu \quad \text{--- (3)}$

Adding equation (1) and equation (2) gives: $(x + y + z) + (x + 2y + 5z) = 6 + 10$ $2x + 3y + 6z = 16 \quad \text{--- (4)}$

For the system to have infinitely many solutions, equation (3) must be identical to equation (4). Comparing the coefficients and constant terms of equation (3) and equation (4): $\lambda = 6$ $\mu = 16$

Thus, the value of $\lambda + \mu$ is: $\lambda + \mu = 6 + 16 = 22$

Hence, the correct option is C.