What is the number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the sides of the octagon?
(A). 24
(B). 52
(C). 48
(D). 16

Question

What is the number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the sides of the octagon?(A). 24(B). 52(C). 48(D). 16

Vedantu Content Team · Accepted Answer

Hint- In this question, first calculate the total number of triangles formed and then subtract from them the number of triangles having one and two sides common with sides of the octagon. This will leave the triangles which have only vertices common with the octagon.Complete step-by-step solution -Total number of Triangles formed equals the total combinations so formed by using any 3 vertices from the 8 vertices i.e. $ = {}^8{C_3}$Now using, ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$${}^8{C_3} = \dfrac{{8!}}{{3!(8 - 3)!}} = \dfrac{{8!}}{{3!5!}} = \dfrac{{8 \times 7 \times 6 \times 5!}}{{3 \times 2 \times 1 \times 5!}} = 56$Now, number of triangles formed with one side common with octagon $ = 8 \times 4 = 32$Here in the Figure (1) below we can see that the side AB of octagon is common for the triangles $\Delta ACB,\Delta ADB,\Delta AEB,\Delta AFB$ . Now for one side 4 triangles are formed so for 8 sides total triangles so formed are $ = 8 \times 4 $ = 32

Figure (1)Now, number of triangles formed with two sides common with octagon = 8 and we can see that 8 triangles i.e. $\Delta AHB,\Delta HCA,\Delta CDH,\Delta DEC,\Delta EFD,\Delta FGE,\Delta GBF,\Delta BAG$ in Figure (2)

Figure (2)So, number of triangles whose vertices are vertices of octagon but none of sides happen to come from sides of octagon = Total number of Triangles formed equals the total combinations so formed by using any 3 vertices from the 8 vertices - number of triangles formed with one side common with octagon - number of triangles formed with two sides common with octagon = 56 – 32 – 8 = 16Hence, number of triangles whose vertices are at the vertices of octagon but none of sides happen to come from sides of octagon = 16$\therefore $ Option D. 16 is the correct answer.Note- For these types of questions use the permutation and combination techniques to find the formula to directly analyze the problem and deduce the trick to find the total number of triangles formed. Also, keep in mind that ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$ where n represents the number of items, and r represents the number of items being chosen at a time.