6.3 Q-2

Exercise 6.3 — Question 2

This exercise covers combinations, their symmetry property, and geometric applications such as counting diagonals in polygons.

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Key Formula

The number of combinations of $n$ distinct objects taken $r$ at a time is:

${}^n{C}_r=\frac{n!}{r!(n-r)!}$

Symmetry Property

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

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Worked Examples

Example 1 — Solving for $n$ using the symmetry property

Problem: If ${}^n{C}_{15}={}^n{C}_7$, find $n$.

Solution:

Using the symmetry property: since $15\ne 7$, we apply $p+q=n$: $n=15+7=22$

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Example 2 — Number of diagonals in a polygon

Problem: Find the number of diagonals in a polygon with $n$ sides.

Solution:

• Total line segments joining any 2 vertices $={}^n{C}_2=\frac{n(n-1)}{2}$
• Subtract the $n$ sides of the polygon:

$\text{Diagonals}={}^n{C}_2-n=\frac{n(n-1)}{2}-n=\frac{n(n-3)}{2}$

Example: For a hexagon ($n=6$): $\text{Diagonals}=\frac{6\times 3}{2}=9$

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Example 3 — Committee selection

Problem: In how many ways can a committee of 3 be chosen from 8 people?

Solution:

${}^8{C}_3=\frac{8!}{3!\cdot 5!}=\frac{8\times 7\times 6}{3\times 2\times 1}=56\text{ ways}$

Example 4 — Selection with a constraint

Problem: In how many ways can a team of 4 be chosen from 7 people if one specific person must always be included?

Solution:

Since one person is fixed, we choose the remaining 3 from the other 6: ${}^6{C}_3=\frac{6!}{3!\cdot 3!}=20\text{ ways}$

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Real-World Applications