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4 points out of 11 points in a plane are...

4 points out of 11 points in a plane are collinear. Number of different triangles that can be drawn by joining them, is a. 165 b. 161 c. 152 d. 159

A

165

B

161

C

152

D

159

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The correct Answer is:
To solve the problem of finding the number of different triangles that can be formed by joining 11 points in a plane, where 4 of these points are collinear, we can follow these steps: ### Step-by-Step Solution: 1. **Identify the Total Points and Collinear Points**: - We have a total of 11 points. - Out of these, 4 points are collinear. - Therefore, the number of non-collinear points is \( 11 - 4 = 7 \). 2. **Calculate the Number of Triangles**: - A triangle can be formed by selecting 3 points from the available points. However, we need to consider the cases based on the collinearity of the points. 3. **Case 1: All 3 Points from Non-Collinear Points**: - We can select 3 points from the 7 non-collinear points. - The number of ways to choose 3 points from 7 is given by the combination formula \( \binom{n}{r} \): \[ \text{Number of triangles} = \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \] 4. **Case 2: 1 Collinear Point and 2 Non-Collinear Points**: - We can select 1 point from the 4 collinear points and 2 points from the 7 non-collinear points. - The number of ways to do this is: \[ \text{Number of triangles} = \binom{4}{1} \times \binom{7}{2} = 4 \times \frac{7!}{2!(7-2)!} = 4 \times \frac{7 \times 6}{2 \times 1} = 4 \times 21 = 84 \] 5. **Case 3: 2 Collinear Points and 1 Non-Collinear Point**: - We can select 2 points from the 4 collinear points and 1 point from the 7 non-collinear points. - The number of ways to do this is: \[ \text{Number of triangles} = \binom{4}{2} \times \binom{7}{1} = 6 \times 7 = 42 \] 6. **Total Number of Triangles**: - Now, we sum the number of triangles from all cases: \[ \text{Total} = 35 + 84 + 42 = 161 \] ### Final Answer: The total number of different triangles that can be drawn by joining these points is **161**.
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