Pedro decided to take all his children, boys and girls, for ice cream at the Ice Cream Math parlor. At the parlor, there are 12 different ice cream flavors, and each child ordered a combo with 3 scoops of ice cream. After leaving the parlor, Pedro realized that exactly two scoops of each available flavor were ordered in total.

a) Knowing that Pedro did not have ice cream, what is the total number of his children (girls and boys)?

b) Pedro looked more closely at the flavors each one ordered and noticed that no flavor was chosen by both a boy and a girl, that is, if a boy chose a flavor, no girl chose that same flavor. Knowing that at least one of his children is a boy and that he has more daughters than sons, determine the number of his daughters.
Solution

a) Let $n$ be the number of Pedro's children. In total, $3 n$ scoops of ice cream were ordered. Since each of the 12 flavors was ordered twice, we have $3 n=2 \cdot 12$, that is, $n=8$. Therefore, Pedro has 8 children.

b) Let $x$ be the number of boys and $y$ be the number of girls. From the previous item, we know that $x+y=8$. Since there is at least one child and there are more girls than boys, we know that: $0<x<y$. Given that no flavor was ordered simultaneously by boys and girls, the flavors can be separated into boys' flavors and girls' flavors. Thus, $3 x$ is equal to twice the number of boys' flavors, as exactly two scoops of each flavor were ordered and no girl could order one of the boys' flavors. Consequently, $x$ is even. Similarly, $y$ is also even. Since $0<x<y$ and $x+y=8$, we can conclude that $x=2$ and $y=6$. Therefore, Pedro has 6 daughters.