There is a standard result regarding this :
If the graph is connected simple planar graph and it contains no triangles , then no of edges e should satisfy :
e <= 2n - 4
Hence given n = 10 ,
The maximum no of edges in such graph = 2*10 - 4
= 16
EDIT : Proof of the given result :
Let we have f faces and as we are given that no triangle should be present in the graph , hence minimum no of edges will be 4 in each face..Also each edge will be counted twice as it will serve as an edge for 2 faces..Hence we have :
e >= 4f/2 = 2f
==> f <= e/2 .........(1)
Now as the graph is planar so Eulers theorem holds ..
We know , if a graph is connected and planar ,
f = e - n + 2 ..........(2)
From (1) , we have :
f <= e/2
==> e - n + 2 <= e/2
==> e - e/2 <= n - 2
==> e <= 2n - 4