Why didn't we use permutation formula for solving after we made the table for number of children in last two families?

I understood till the last part where we wrote say 1 and 5 , 2 and 4… till 5 and 1.

However, my doubt is why didn’t we use permutation formula for solving after we made the table for number of children in last two families?

Like (6P1* 6P5) + (6P26P4)+ (6P36P3) + (6P46P2) + (6P56P1) ??? Why not like this?

Why did we simply count the number of possibilities as 5 ?