Problem 2.3. Suppose you flip 50 fair coins. (a) How many possible outcomes (microstates) are there? (b) How many ways are there of getting exactly 25 heads and 25 tails? (c) What is the probability of getting exactly 25 heads and 25 tails? (d) What is the probability of getting exactly 30 heads and 20 tails? (e) What is the probability of getting exactly 40 heads and 10 tails? (f) What is the probability of getting 50 heads and no tails? (g) Plot a graph of the probability of getting \( n \) heads, as a function of \( n \)

Perform the row operations \( R_{2}=-3 r_{1}+r_{2} \) and \( R_{3}=9 r_{1}+r_{3} \) on the original matrix given in the problem, and enter the resulting matrix below. (Simplify your answer.)

3. Determine o valor real de \( y \) para que as expressões \( \frac{3 y}{y-4} \) e \( 3+\frac{2}{y} \) sejam iguais, sabendo que \( y \neq 0 \) e \( y \neq 4 \)

14. En la expresión \( 5=\pi r^{2}, r^{2}= \)

5. What is the image of point \( (0,-5) \) after translation by \( (2,7) \) ?

ones: tipo examen Resuelva la ecuación diferencial \( y^{(4)}-3 y^{\prime \prime \prime}+3 y^{\prime \prime}-y^{\prime}=0 \quad y(0)=0, y^{\prime}(0)=0, y^{\prime \prime}(0)=1, y^{\prime \prime \prime}(0)=1 \) Respuesta: \( y=2-2 e^{x}+2 x e^{x}-\frac{1}{2} x^{2} e^{x} \)