1. Recall the periodic trend for first ionization enthalpy

   First ionization enthalpy generally:

   • increases across a period from left to right,
   • decreases down a group.

2. Compare the given elements

   The elements are:

   • $\mathrm{Ca}$ : Group 2, Period 4
   • $\mathrm{Ba}$ : Group 2, Period 6
   • $\mathrm{S}$ : Group 16, Period 3
   • $\mathrm{Se}$ : Group 16, Period 4
   • $\mathrm{Ar}$ : Group 18, Period 3

3. Compare $\mathrm{Ba}$ and $\mathrm{Ca}$

   Both belong to Group 2. Down the group, ionization enthalpy decreases. Therefore, $\mathrm{Ba} < \mathrm{Ca}$

4. Compare $\mathrm{Se}$ and $\mathrm{S}$

   Both belong to Group 16. Down the group, ionization enthalpy decreases. Therefore, $\mathrm{Se} < \mathrm{S}$

5. Compare Group 2 metals with Group 16 nonmetals

   Nonmetals generally have much higher first ionization enthalpy than alkaline earth metals in nearby periods. Hence, $\mathrm{Ca} < \mathrm{Se}, \qquad \mathrm{Ba} < \mathrm{Se}$ and also $\mathrm{Ca} < \mathrm{S}, \qquad \mathrm{Ba} < \mathrm{S}$

6. Compare with argon

   $\mathrm{Ar}$ is a noble gas and has a very high first ionization enthalpy due to its stable closed-shell configuration. Thus, among all given elements, $\mathrm{Ar}$ has the highest first ionization enthalpy.

7. Arrange in increasing order

   Combining all comparisons: $\mathrm{Ba} < \mathrm{Ca} < \mathrm{Se} < \mathrm{S} < \mathrm{Ar}$

8. Match with the options

   This corresponds to:

   • Option B: $\mathrm{Ba < Ca < Se < S < Ar}$

Therefore, the correct answer is B.