Question

How many triangles can be formed by joining the vertices of a hexagon?
A. 31
B. 25
C. 20
D. 60

Answer 1

Hint: As we know that hexagon has $$6$$ vertices means $$6$$ points and the triangle has $$3$$ points means a triangle need $$3$$ vertices to be formed. Then, the numbers of triangles that can be formed by joining the vertices of a hexagon can be calculated by applying the concept of combination.

Complete step by step solution:

The number of vertices in a hexagon is $$6$$.
The number of vertices in a triangle is $$3$$.
That means to make a triangle we need $$3$$ vertices.
So,$$6$$is the total number of vertices in hexagons, and out of those 6 vertices, 3 vertices that being chosen at a time to make a triangle.
By applying the formula of combination:
$${}^n{C}_r={\displaystyle \frac{n!}{r!\times \left(n-r\right)!}}$$ where, n is the total numbers of objects and r is the number of objects to be chosen as a time.
Therefore,
$${}^6{C}_3={\displaystyle \frac{6!}{3!\times \left(6-3\right)!}}={\displaystyle \frac{6!}{3!\times \left(3\right)!}}$$
By using the values of factorial terms:
We have, $$6!$$ can be written as $$6\times 5\times 4\times 3\times 2\times 1$$ and $$3!$$ can be written as $$3\times 2\times 1.$$
Replace $$6!$$ = $$6\times 5\times 4\times 3\times 2\times 1$$ and $$3!$$= $$3\times 2\times 1$$ in the following equation,
$$\begin{gathered} {}^6{C}_3={\displaystyle \frac{6!}{3!\times \left(3\right)!}} \\ ={\displaystyle \frac{6\times 5\times 4\times 3\times 2\times 1}{3\times 2\times 1\left(3\times 2\times 1\right)}}={\displaystyle \frac{120}{6}}=20 \end{gathered}$$
$\therefore$ The number of triangles that can be formed by joining the vertices of a hexagon is $$20$$. Hence, option (C) is correct.

Note:
These types of questions always use a combination concept for solving the problem.
Some important definitions you should know
Vertices mean the point where $$2$$ or more lines meet.
Combination: Any of the ways we can combine things when the order does not matter.
Combination formula: The formula of combination is $${}^n{C}_r={\displaystyle \frac{n!}{r!\times \left(n-r\right)!}}$$, where $$n$$ represents the number of items and $$r$$ represents the number of the items that being chosen at a time.
The factorial function$$(!)$$: The factorial function means to multiply all whole numbers from the chosen numbers down to $$1$$.
The representation of factorial is $$n!$$.
Alternatively, you can draw the figure and count the triangles but it becomes complicated.