Every student in the senior class is taking history or science. There are $200$ seniors in the class. If there are $126$ seniors taking history and $129$ seniors taking science, how many students are taking both history and science?
I want to find the number of seniors taking both history and science, so I will use the formula for the intersection of two sets: $|A \cap B| = |A| + |B| - |A \cup B|$. In this formula, $|A|$ is the number of elements in set $A$, $|B|$ is the number of elements in set $B$, and $|A \cup B|$ is the number of elements in the union of both sets. In this problem, I can let $A$ be the set of seniors taking history and $B$ be the set of seniors taking science. Then, I have $|A| = 126$, $|B| = 129$, and $|A \cup B| = 200$, since every senior is taking either history or science or both. Plugging these values into the formula, I get $|A \cap B| = 126 + 129 - 200 = 55$. Therefore, there are $55$ seniors taking both history and science. 