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

vans = \( \frac{30}{6} \) = 5

The equation for this sequence is:

a_n = a_n-1 + 5

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 5
a₆ = 21 + 5
a₆ = 26

First, solve for \( \left|a + 3\right| \):

\( \left|a + 3\right| \) + 1 = -8
\( \left|a + 3\right| \) = -8 - 1
\( \left|a + 3\right| \) = -9

The value inside the absolute value brackets can be either positive or negative so (a + 3) must equal - 9 or --9 for \( \left|a + 3\right| \) to equal -9:

So, a = 6 or a = -12.

By buying two shirts, Charlie will save $20 x \( \frac{50}{100} \) = \( \frac{$20 x 50}{100} \) = \( \frac{$1000}{100} \) = $10.00 on the second shirt.

So, his total cost will be
$20.00 + ($20.00 - $10.00)
$20.00 + $10.00
$30.00

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:

\( \frac{2!}{4!} \)
\( \frac{2 \times 1}{4 \times 3 \times 2 \times 1} \)
\( \frac{1}{4 \times 3} \)
\( \frac{1}{12} \)