Answer :
To solve the problem of determining how many pentagons can be formed by connecting five points from [tex]\( n \)[/tex] points on a circle, we need to consider the concept of combinations.
Here's a step-by-step explanation:
1. Understand the Problem: We have [tex]\( n \)[/tex] points on a circle, and we want to form pentagons by selecting any 5 of these points. A pentagon requires exactly 5 points.
2. Use Combinations: Since the order of selecting the points does not matter, we use combinations rather than permutations. The number of ways to choose 5 points from [tex]\( n \)[/tex] points is given by the combination formula:
[tex]\[
\binom{n}{5} = \frac{n!}{5!(n-5)!}
\][/tex]
This formula calculates the number of ways to choose 5 points from [tex]\( n \)[/tex] points without regard to order.
3. Interpret the Result: The result of the combination formula [tex]\(\binom{n}{5}\)[/tex] is the number of different pentagons that can be formed. Each set of 5 points corresponds to a unique pentagon because they define the vertices of the pentagon on the circle.
4. Conclusion: Therefore, the number of pentagons you can draw by connecting any five of these [tex]\( n \)[/tex] points is [tex]\(\binom{n}{5}\)[/tex] pentagons.
This method guarantees that each selection of 5 points is counted once, providing the total number of unique pentagons possible.
Here's a step-by-step explanation:
1. Understand the Problem: We have [tex]\( n \)[/tex] points on a circle, and we want to form pentagons by selecting any 5 of these points. A pentagon requires exactly 5 points.
2. Use Combinations: Since the order of selecting the points does not matter, we use combinations rather than permutations. The number of ways to choose 5 points from [tex]\( n \)[/tex] points is given by the combination formula:
[tex]\[
\binom{n}{5} = \frac{n!}{5!(n-5)!}
\][/tex]
This formula calculates the number of ways to choose 5 points from [tex]\( n \)[/tex] points without regard to order.
3. Interpret the Result: The result of the combination formula [tex]\(\binom{n}{5}\)[/tex] is the number of different pentagons that can be formed. Each set of 5 points corresponds to a unique pentagon because they define the vertices of the pentagon on the circle.
4. Conclusion: Therefore, the number of pentagons you can draw by connecting any five of these [tex]\( n \)[/tex] points is [tex]\(\binom{n}{5}\)[/tex] pentagons.
This method guarantees that each selection of 5 points is counted once, providing the total number of unique pentagons possible.
