Three-fourths of the students in Mr. Shearer's class have brown hair and six-sevenths of his students are right-handed. If Mr. Shearer's class has 28 students, what is the smallest possible number of students that could be both right-handed and have brown hair?
\(|A \cup B| = |A|+|B| - |A \cap B|\)
let A be brown haired students, and B be right handed students
\(|A|=\dfrac 3 4 \cdot 28 = 21 \\ |B| = \dfrac 6 7 \cdot 28 = 24 \\ |A \cup B| \leq 28 \\ 21 + 24 - |A \cap B| \leq 28 \\ 17 \leq |A \cap B|\)
so the smallest number of students that have both brown hair and are right handed is 17