6.3 Q-11

Exercise 6.3 — Question 11

Problem

Using the symmetry property of combinations, solve for $n$ if ${}^n{C}_{15}={}^n{C}_7$.

Also, find the number of diagonals in a polygon with $n$ sides.

---

Key Property: Symmetry of Combinations

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

This follows directly from the formula ${}^n{C}_r=\frac{n!}{r!(n-r)!}$, since ${}^n{C}_r={}^n{C}_{n-r}$.

---

Part (i): Solve ${}^n{C}_{15}={}^n{C}_7$

Since $15\ne 7$, we use the second condition: $p+q=n⟹15+7=n⟹\boxed{n=22}$

---

Part (ii): Number of Diagonals in an $n$-sided Polygon

A polygon with $n$ vertices has ${}^n{C}_2$ total line segments connecting any two vertices.

Out of these, $n$ segments are the sides of the polygon.

Therefore, the number of diagonals is: ${}^n{C}_2-n=\frac{n(n-1)}{2}-n=\frac{n(n-1)-2n}{2}=\frac{n^2-3n}{2}=\boxed{\frac{n(n-3)}{2}}$

Example: A hexagon ($n=6$) has $\frac{6\times 3}{2}=9$ diagonals.

---