Try this beautiful problem from combinatorics PRMO 2019 based on Regular polygon
In how many ways can a pair of parallel diagonals of a regular polygon of
But try the problem first...
PRMO-2019, Problem 15
Pre College Mathematics
The above diagram is a diagram of Regular Polygon .we have to draw the diagonals as shown in above.we joined the diagonals such that all the diagonals will be parallel
Can you now finish the problem ..........
If we joined the diagonals (shown in Fig. 1), i.e \((P_3 \to P_10)\),\((P_4\to P_9)\),\((P_5 \to P_8)\) then then we have 3 diagonals.so we have 5\(4 \choose 2\) ways=\(15\) ways.
If we joined the diagonals (shown in Fig.2), i.e \((P_1 \to P_3)\),\((P_10\to P_4)\),\((P_9\to P_5)\),\((P_8\to P_6)\)then we have \(4\)diagonals.so we have 5\(3 \choose 2\) ways=\(30\) ways.
Therefore total numbers of ways that can a pair of parallel diagonals of a regular polygon of \(10\) sides be selected is \(15+30=45\)