The number of pairs of diagonals of a regular polygon of `10` sides that are parallel are

Explanation

Fix a vertex, say A, as shown in the first figure.

There are 4 parallel diagonals.
There are 5 such sets since the vertices of the polygon can be paired into 5 sets.
In each set, there are ` .^(4)C_(2)` pairs, i.e., a total of `.^(4)C_(2)xx5=30` pairs of diagonals.
Now, fix a side AB as shown in the second figure.
There are 3 parallel diagonals and 5 such sets.
So, total number of pairs `= .^(3)C_(2)xx5=15`
Hence, total number of required diagonals=30+15=45