【Example 2】As shown in Figure $1, \triangle A B C$ has $n_{1}, n_{2}, n_{3}$ points on its sides (excluding points $A, B, C$). By selecting one point from each of the three sides as the vertices of a triangle, how many different triangles can be formed?
Solution: Since we need to select one point from the $n_{1}$ points on side $AB$, one point from the $n_{2}$ points on side $BC$, and one point from the $n_{3}$ points on side $AC$ as the vertices of the triangle, this is a 3-step selection problem. According to the multiplication principle: we can form $n_{1} n_{2} n_{3}$ different triangles.