A ratio of 5:1 means that there are 5 home fans for every one visiting fan. So, of every 6 fans, 5 are home fans and \( \frac{5}{6} \) of every fan in the stadium is a home fan:

34,000 fans x \( \frac{5}{6} \) = \( \frac{170000}{6} \) = 28,333 fans.

First, solve for \( \left|z - 3\right| \):

\( \left|z - 3\right| \) - 6 = 0
\( \left|z - 3\right| \) = 0 + 6
\( \left|z - 3\right| \) = 6

The value inside the absolute value brackets can be either positive or negative so (z - 3) must equal + 6 or -6 for \( \left|z - 3\right| \) to equal 6:

So, z = -3 or z = 9.

The factors of a number are all positive integers that divide evenly into the number. The factors of 16 are 1, 2, 4, 8, 16.

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{2!}{3!} \)
\( \frac{2 \times 1}{3 \times 2 \times 1} \)
\( \frac{1}{3} \)
\( \frac{1}{3} \)

To divide fractions, invert the second fraction and then multiply:

\( \frac{3}{5} \) ÷ \( \frac{1}{5} \) = \( \frac{3}{5} \) x \( \frac{5}{1} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{3}{5} \) x \( \frac{5}{1} \) = \( \frac{3 x 5}{5 x 1} \) = \( \frac{15}{5} \) = 3