The total number of sides of two polygons is $17$ and total number of diagonals is $53$. The numbers of sides of the polygons are —

A polygon with $n$ sides has $\dfrac{n(n-3)}{2}$ diagonals. For sides $6$ and $11$: $\dfrac{6\cdot3}{2} + \dfrac{11\cdot8}{2} = 9 + 44 = 53$, and $6 + 11 = 17$, satisfying both conditions.