\(\mathrm{ABCD}\) is a convex quadrilateral. \(3,4,5\) and 6 points are marked on the sides \(\mathrm{AB}, \mathrm{BC}, \mathrm{CD}\) and \(\mathrm{DA}\) resp. The number of triangles with vertices on different sides are (a) 270 (b) 220 (c) 282 (d) 342

A convex quadrilateral is a simple four-sided figure with all its interior angles less than 180 degrees, ensuring that the line segment connecting any two points within the quadrilateral does not extend beyond its sides. This property becomes crucial in geometric combinatorics when we examine the arrangement of points and the formation of triangles.
In our exercise, the quadrilateral ABCD hosts points on its sides that are candidates for the vertices of possible triangles. The convex nature implies no triangle formed will have a vertex that 'dips' inside the quadrilateral. This simplicity offers a clear pathway to calculating triangular combinations – a key aspect of understanding the relationship between quadrilaterals and combinatorics.

Combinations and permutations are foundational concepts in combinatorics, the branch of mathematics focused on counting. While permutations are about arranging items where the order is important, combinations concern themselves with selections where the order does not matter.
In the context of our problem, we are dealing with combinations—specifically, selecting points from different sides of the quadrilateral to form a triangle. The formula for combinations is given by
\[\begin{equation}C(n, k) = \frac{n!}{k!(n-k)!}\br\br\bb \end{equation}\],where 'n' represents the total number of items, 'k' represents the number of items to select, and '!' denotes factorial, which is the product of all positive integers up to that number. The step-by-step solution uses this formula to calculate the number of triangular combinations from points on the sides of a convex quadrilateral, clearly illustrating the power of combinatorics in geometry.

Triangular combinations refer to the ways we can select points to form triangles, subject to certain conditions. In our example, the condition is that each triangle's vertices must lie on different sides of the convex quadrilateral.
To find the number of triangular combinations, we use the idea of combinations from different sets of points. It requires finding all possible groups of three sides from the quadrilateral and then calculating the product of the number of points on each chosen side—an application of the fundamental counting principle.
\[\begin{equation}Triangular \thinspace Combinations = \thinspace Product \thinspace of \thinspace Points \thinspace on \thinspace Chosen \thinspace Sides\br\br\bb \end{equation}\].The problem breaks this down into separate calculations for each group of sides, ensuring clarity in understanding. This approach effectively communicates how combinatorics principles can unravel complex geometric configurations, tying together the importance of the convex quadrilateral and combinations in determining the number of possible triangles.