Suppose we have a group of 5 children and randomly select 2 from the group without replacement. how many possible unique samples of size 2 are possible if the order of selection does not matter?

If the order of selection is not important we can create a combination of 2 children out 5 children. Let us suppose we have 5 children 1,2,3,4, and 5. Number 1 child can create four combination with other four children as 12, 13, 14, and 15. If second child, is making a combination with other children he has one combination common with child 1 combination that is 12. So by reducing common pair we have 4 pairs from child 1, 3 pairs from child 2, 2 pairs from child 3, and 1 from child 4. There is no unique combination left for child 5. In total we have 4+3+2+1=10 or 2 combination 5 i.e. 2*5=10 unique samples of size 2 children.