discrete math functions pigeonhole princple How many multiples

Proving the double differential of $z = -z$ implies ${\displaystyle z=\sin x}$
${\displaystyle \frac{d^{2}z}{{ dx }^2}=-z}$
implies z is of the form ${\displaystyle a\sin x+b\cos x}$. Is there a proof for the same. I was trying to arrive at the desired function but couldn't understand how to get these trigonometric functions in the equations by integration. Does it require the use of taylor polynomial expansion of ${\displaystyle \sin x\text{or}\cos x}$?

In a family of 6, what is the probability of getting 1 boy and 5 girls

find all values of k sor which they given matrix augmented matrix to a consistent linear system

We wish to estimate what percent of adult residents in a certain county are parents. Out of 600 adult residents sampled, 336 had kids. Based on this, construct a 90% confidence interval for the proportion p of adult residents who are parents in this county.

Random-digit-dialing telephone surveys used to exclude cell phone numbers. If the opinions of people who have only cell phones differed from those of people who have landline service, the poll results may not represent the entire adult population. The Pew Research Center interviewed separate random samples of cell-only and landline telephone users who were less than 30 years old and asked them to describe their political party affiliation.
$\text{□}$ or goodness of fit, homogeneity or independence?