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There are 10 points in a planą, out of these 6 are collnear. IfN is the number of trangles formed by joining these points, then?
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There are 10 points in a planą, out of these 6 are collnear. IfN is th...
To determine the number of triangles that can be formed from 10 points in a plane, where 6 points are collinear, we need to consider the conditions under which a triangle can be formed.
Understanding Triangle Formation
- A triangle requires 3 non-collinear points.
- Collinear points cannot form a triangle.
Calculating Total Combinations
- The total number of ways to choose 3 points from 10 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of points and \( r \) is the number of points to choose.
- For our case:
\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
Excluding Collinear Points
- We need to exclude triangles formed by the 6 collinear points.
- The number of ways to choose 3 points from these 6 collinear points is:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
Calculating Valid Triangles
- To find the valid triangles, we subtract the collinear combinations from the total combinations:
\[ \text{Valid triangles} = \binom{10}{3} - \binom{6}{3} = 120 - 20 = 100 \]
Conclusion
- The total number of triangles that can be formed by joining these points is **100**.
Understanding Triangle Formation
- A triangle requires 3 non-collinear points.
- Collinear points cannot form a triangle.
Calculating Total Combinations
- The total number of ways to choose 3 points from 10 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of points and \( r \) is the number of points to choose.
- For our case:
\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
Excluding Collinear Points
- We need to exclude triangles formed by the 6 collinear points.
- The number of ways to choose 3 points from these 6 collinear points is:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
Calculating Valid Triangles
- To find the valid triangles, we subtract the collinear combinations from the total combinations:
\[ \text{Valid triangles} = \binom{10}{3} - \binom{6}{3} = 120 - 20 = 100 \]
Conclusion
- The total number of triangles that can be formed by joining these points is **100**.
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There are 10 points in a planą, out of these 6 are collnear. IfN is the number of trangles formed by joining these points, then?
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There are 10 points in a planą, out of these 6 are collnear. IfN is the number of trangles formed by joining these points, then? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about There are 10 points in a planą, out of these 6 are collnear. IfN is the number of trangles formed by joining these points, then? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There are 10 points in a planą, out of these 6 are collnear. IfN is the number of trangles formed by joining these points, then?.
There are 10 points in a planą, out of these 6 are collnear. IfN is the number of trangles formed by joining these points, then? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about There are 10 points in a planą, out of these 6 are collnear. IfN is the number of trangles formed by joining these points, then? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There are 10 points in a planą, out of these 6 are collnear. IfN is the number of trangles formed by joining these points, then?.
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