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Which of the following statements is false?

A An octagon has $20$ diagonals

B A hexagon has $9$ diagonals

C A hexagon has $4$ more diagonals than a pentagon

D A pentagon has the same number of diagonals as it has sides

E A quadrilateral has twice as many diagonals as it has sides

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.