Question

At a certain university 523 of the seniors are history majors or math majors(or both).There are 100 senior math majors, and 33 seniors are majoring in both history and math. How many seniors are majoring in history?

Answer 1

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1)
Inclusion-Exclusion Formula:
If A and B are any two finite sets, then \(\displaystyle{\left|{{A}\cup{B}}\right|}={\left|{{A}}\right|}+{\left|{{B}}\right|}-{\left|{{A}\cap{B}}\right|}\).
Let H, M be the sets of seniors whose major is History and Math respectively.
It is given that 523 seniors are history majors or math majors or both.
Thus, \(\displaystyle{\left|{{H}\cup{M}}\right|}={523}\).
It is given that 100 seniors are math majors.
Thus, \(\displaystyle{\left|{{M}}\right|}={100}\).
The number of seniors majoring in math and history are 33.
Thus, \(\displaystyle{\left|{{H}\cap{M}}\right|}={33}\).
Step 2
Evaluate the number of seniors majoring in History using Inclusion-Exclusion principle as follows.
\(\displaystyle{\left|{{H}\cup{M}}\right|}={\left|{{H}}\right|}+{\left|{{M}}\right|}-{\left|{{H}\cap{M}}\right|}\)
\(\displaystyle{523}={\left|{{H}}\right|}+{100}-{33}\)
\(\displaystyle{523}={\left|{{H}}\right|}+{67}\)
\(\displaystyle{\left|{{H}}\right|}={523}-{67}\)
\(\displaystyle{\left|{{H}}\right|}={456}\)
Therefore, the number of seniors majoring in History is 456.