Problem 4 How many different passwords of length eight (repetition allowed) can one make such that: - The password is made out of big letters, small letters, numbers, and special characters. - There are 27 big letters, 27 small letters, 10 numbers ja 36 special characters available. - The password has to have at least one big letter, one number or one special character. (Any of these conditions suffices.)

\( \int \frac { 4 \ln ( x ) + 3 } { x ( 2 \ln ( x ) + 1 ) } d x \)

Solve for \( f \) \( \begin{array}{l}3(6-f)-4=3 f-4 \\ f=\square\end{array} \)

Simplify the following to a single trigonometric function, without the use of a calculator. \( \frac{\cos \left(90^{\circ}-\alpha\right) \sin \left(-\alpha-540^{\circ}\right)}{\tan 225^{\circ}+\sin \alpha \cdot \sin \left(180^{\circ}+\alpha\right)} \) Siven: \( 1-\cos \theta=2 \sin ^{2} \theta \) 5.4.1 Show that the equation can be written as: \( (2 \cos \theta+1)(\cos \theta-1)=0 \).

Erectincoun 9. El valor numérico de las siguientes expre- siones es 0 para algunos números. Indica cuáles son. \( \begin{array}{ll}\text { a) } x^{2}-64 & \text { b) } x^{3}-1000 \\ \text { c) } x^{2}+25 & \text { d) } x^{3}+64\end{array} \)

2. Expresen en minutos. a. \( 360^{\prime \prime}=\square \) b. \( 45^{\circ} 120^{\prime \prime}=\square \)