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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.

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.