6.3 Q-12

Exercise 6.3 — Question 12

Problem Types Covered

This question set covers two key applications of combinations:

1. Symmetry property of combinations — solving equations of the form ${}^n{C}_p={}^n{C}_q$
2. Counting diagonals in a polygon — a classic real-world application of combinations

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Symmetry Property of Combinations

Recall the combination formula: ${}^n{C}_r=\frac{n!}{r!(n-r)!}$

Key Property: If ${}^n{C}_p={}^n{C}_q$, then either: $p=q\text{or}p+q=n$

This follows directly from the identity ${}^n{C}_r={}^n{C}_{n-r}$.

Example: Solve ${}^n{C}_{15}={}^n{C}_7$.

Since $15\ne 7$, we use $p+q=n$: $n=15+7=22$

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Diagonals of a Polygon

An $n$-sided polygon has $n$ vertices. A diagonal connects two non-adjacent vertices.

Step 1: Total line segments connecting any 2 vertices: ${}^n{C}_2=\frac{n(n-1)}{2}$

Step 2: Subtract the $n$ sides of the polygon (which are not diagonals): $\text{Number of diagonals}={}^n{C}_2-n=\frac{n(n-1)}{2}-n=\frac{n(n-3)}{2}$

Example: Find the number of diagonals in a hexagon ($n=6$): $\text{Diagonals}=\frac{6(6-3)}{2}=\frac{6\times 3}{2}=9$

Example: Find the number of diagonals in a decagon ($n=10$): $\text{Diagonals}=\frac{10(10-3)}{2}=\frac{10\times 7}{2}=35$