Three black holes, with masses of $M_1,M_2,$ and $M_3,$ are merged.

a$.$ Show that the minimum mass of the resulting black hole does not depend on the order in which they are merged. In other words, show that merging black holes $1$ and $2$ first and then adding black hole $3,$ has the same minimum mass as merging black holes $2$ and $3$ first and then adding black hole $1.$

b$.$ What is the minimum possible mass of the final black hole after all three have merged?

Instead, $N$ identical black holes, each with mass $\frac{M_T}{N}$ are merged.

c$.$ What is the minimum possible mass of the resulting black hole?

d$.$ What is the limit of this minimum mass as $N?$