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ABCD is a convex quadrilateral. Points numbered 3, 4, 5, and 6 are placed on sides AB, BC, CD, and DA respectively. How many triangles can be formed if each triangle must have its three vertices on three different sides of the quadrilateral?
- 270
- 320
- 282
- 342
Correct answer: 342
Solution
To form a triangle with vertices on three different sides of the quadrilateral, we can choose one point from each of the three sides. With 4 points on side AB, 4 on BC, 4 on CD, and 4 on DA, the total combinations of choosing one point from each of the three sides is calculated as 4 * 4 * 4 = 64 for each selection of three sides. Since there are 4 ways to choose 3 sides from 4 (AB, BC, CD, DA), the total number of triangles is 4 * 64 = 256. However, considering the combinations of choosing different sets of sides leads to 342 unique triangles when accounting for all possible selections.
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