The minimum possible sum of all the children's ages is 44.

Here's the reasoning:
1. We know the youngest child is 4 years old and the oldest child is 16 years old.
2. To minimize the sum of their ages, we need to maximize the age differences between consecutive siblings while still maintaining a minimum gap of 2 years.
3. Working backward from the oldest sibling (16), we can assign ages as follows: 16, 14, 12, 8, and 4.
4. This distribution maintains the minimum 2-year gap between consecutive siblings and minimizes the sum of their ages.
5. Adding up these ages, we get: 4 + 8 + 12 + 14 + 16 = 44.

Hence, the minimum possible sum of all the children's ages is 44.
In a family of 5 children, each child was born in a different year. The youngest child is 4 years old, and the oldest child is 16 years old. If the age difference between any two consecutive siblings is at least 2 years, what is the minimum possible sum of all the children's ages?