A ratio of 4:1 means that there are 4 home fans for every one visiting fan. So, of every 5 fans, 4 are home fans and \( \frac{4}{5} \) of every fan in the stadium is a home fan:

34,000 fans x \( \frac{4}{5} \) = \( \frac{136000}{5} \) = 27,200 fans.

First, solve for \( \left|y - 7\right| \):

\( \left|y - 7\right| \) + 5 = 9
\( \left|y - 7\right| \) = 9 - 5
\( \left|y - 7\right| \) = 4

The value inside the absolute value brackets can be either positive or negative so (y - 7) must equal + 4 or -4 for \( \left|y - 7\right| \) to equal 4:

So, y = 3 or y = 11.

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

\( \frac{1}{6} \) x \( \frac{1}{6} \) = \( \frac{1 x 1}{6 x 6} \) = \( \frac{1}{36} \) = \(\frac{1}{36}\)

Calculate the number of vans needed by dividing the number of people that need transported by the capacity of one van:

vans = \( \frac{48}{8} \) = 6

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:

\( \frac{6!}{4!} \)
\( \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} \)
\( \frac{6 \times 5}{1} \)
\( 6 \times 5 \)
30