The number of triangles having no side common with an \( n \)-sided polygon is given by:
\[ \text{no. of triangles having no side common with a } n \text{-sided polygon} = \binom{n}{1} \times \binom{n-4}{2} \div 3 \]
Substitute \( n = 8 \):
\[ = \binom{8}{1} \times \binom{4}{2} \div 3 \]
\[ = 8 \times 6 \div 3 \]
\[ = 16. \]
Thus, the number of such triangles is 16.
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32