Let Ai where i=1, ldots ldots ldots ldots . .12 are the vertices of a regular dodecagon and G is its centre. Let ' s ' be the number of straight lines that can be formed with these 13 points, ' t ' be number of triangle that can be formed and ' d ' be number of diagonal in a dodecagon. Find the value of (s+t+d).

Question

Let Ai where i=1, ldots ldots ldots ldots . .12 are the vertices of a regular dodecagon and G is its centre. Let ' s ' be the number of straight lines that can be formed with these 13 points, ' t ' be number of triangle that can be formed and ' d ' be number of diagonal in a dodecagon. Find the value of (s+t+d). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

s= 12 C 2=66 t = 13 C 3-6=(13 ⋅ 12 ⋅ 11/6)-6=286-6=280 d = 12 C 2-12=66-12=54 Hence, ( s + t + d )=66+280+54=120+280=400.

s=12C2​=66 t=13C3​−6=613⋅12⋅11​−6=286−6=280 d=12C2​−12=66−12=54 Hence, (s+t+d)=66+280+54=120+280=400.

$s =^{12} C_{2} = 66$
$t =^{13} C_{3} - 6 = \frac{13 \cdot 12 \cdot 11}{6} - 6 = 286 - 6 = 280$
$d =^{12} C_{2} - 12 = 66 - 12 = 54$
Hence, $(s + t + d) = 66 + 280 + 54 = 120 + 280 = 400$ .