Problem Setup
We are given the following information about a group of 35 persons:
- Total number of persons = 35
- Number of young persons = 20
- Number of girls = 18
We need to find the number of young girls in this group.
Finding the Number of Young Girls
Let Y represent the set of young persons and G represent the set of girls.
We are given:
- The size of set Y, $|Y| = 20$
- The size of set G, $|G| = 18$
- The total number of persons in the group is 35.
We are looking for the number of individuals who are both young and girls, which corresponds to the intersection of the sets Y and G, denoted as $|Y \cap G|$.
The number of people who belong to at least one of the sets (Young or Girls) is represented by the union, $|Y \cup G|$. The formula relating union, intersection, and individual set sizes is:
$|Y \cup G| = |Y| + |G| - |Y \cap G|$Substituting the known values:
$|Y \cup G| = 20 + 18 - |Y \cap G|$ $|Y \cup G| = 38 - |Y \cap G|$The total number of people in the union ($|Y \cup G|$) cannot exceed the total number of people in the group, which is 35.
$|Y \cup G| \le 35$Combining the equations, we get:
$38 - |Y \cap G| \le 35$To find the minimum possible value for $|Y \cap G|$, we rearrange the inequality:
$38 - 35 \le |Y \cap G|$ $3 \le |Y \cap G|$This calculation indicates that the minimum number of young girls possible in the group is 3.
Result
Based on the provided numbers, the minimum number of young girls is 3.
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