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The base of the pyramid has $n$ edges, so also has $n$ vertices around the base. This then means that there are $n$ edges around the base (in red) of the pyramid and $n$ that meet at the apex (in black).
Therefore there are $2n$ edges in total.
There are also $n$ faces that meet at the apex of the pyramid, and one more for the base, so a total of $n+1$ faces.
Therefore the difference is $2n - (n+1) = n-1$.
Pyramidal N-gon
Age 14 to 16
ShortChallenge Level 
The base of the pyramid has $n$ edges, so also has $n$ vertices around the base. This then means that there are $n$ edges around the base (in red) of the pyramid and $n$ that meet at the apex (in black).Therefore there are $2n$ edges in total.
There are also $n$ faces that meet at the apex of the pyramid, and one more for the base, so a total of $n+1$ faces.
Therefore the difference is $2n - (n+1) = n-1$.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.
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