Correct Answer - Option 3 : 54
Concept:
The number of ways to select r things out of n things is given by nCr
\(\rm ^nC_r=\frac{n!}{(n-r)!\times(r)!}=\frac{n\times(n-1)\times....(n-r+1)}{r!}\)
Calculation:
A polygon of 12 sides has 12 vertices.
By joining any two of the vertices, we obtain either a side or a diagonal of the polygon.
Number of all straight lines obtained by joining 2 vertices at a time = \(\rm ^{12}C_2=\frac{12\times11}{2\times1}\)
= 66
These straight lines include 12 sides of the polygon.
∴ The number of diagonals of the polygon = 66 - 12
= 54
Hence, option (3) is correct.
The number of diagonals in a polygon of n sides is given by, \(\rm ^nC_2-n\)