Solution

The easiest way is to use the complement.

We want: At least one department receives at least 2 interns.
The opposite is: Every department receives at most 1 intern.

Since there are 4 interns and 4 departments, the only way every department gets at most 1 intern is if each department gets exactly 1 intern.

Total possible assignments: Each of the 4 interns has 4 choices:

\(4^4=256\)

Assign one intern to each department

For the first intern: 4 choices
Second: 3 choices
Third: 2 choices
Fourth: 1 choice

\(4!=24\)

\(P(\text{no department gets 2+})=\frac{24}{256}=\frac{3}{32}\)

Take the complement

\(P(\text{at least one department gets 2+}) =1-\frac{3}{32} =\frac{29}{32}\)

Answer: \(\frac{29}{32}\)
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