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:

\( \frac{6!}{2!} \)
\( \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} \)
\( \frac{6 \times 5 \times 4 \times 3}{1} \)
\( 6 \times 5 \times 4 \times 3 \)
360

There are 4 3-passenger taxis available so that's 4 x 3 = 12 total seats. There are 17 people needing transportation leaving 17 - 12 = 5 who will have to find other transportation.

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for distance:

distance = \( \text{speed} \times \text{time} \)
distance = \( 45mph \times 5h \)
225 miles

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [40, 80] making 40 the smallest multiple 8 and 10 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{8 x 5}{8 x 5} \) - \( \frac{5 x 4}{10 x 4} \)

\( \frac{40}{40} \) - \( \frac{20}{40} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{40 - 20}{40} \) = \( \frac{20}{40} \) = \(\frac{1}{2}\)

A factorial has the form n! and is the product of the integer (n) and all the positive integers below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.