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Try this beautiful problem from combinatorics PRMO 2019 based on Regular polygon

Regular polygon| PRMO | Problem 15

In how many ways can a pair of parallel diagonals of a regular polygon of $10$ sides be selected

• $24$
• $45$
• $34$

Key Concepts

Combinatorics

Regular polygon

geometry

But try the problem first…

Answer:$45$

Source

PRMO-2019, Problem 15

Pre College Mathematics

Try with Hints

First hint

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 ……….

Second Hint

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.

Final Step

Therefore total numbers of ways that can a pair of parallel diagonals of a regular polygon of $10$ sides be selected is $15+30=45$