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:

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

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

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

\( \frac{5 x 5}{9 x 5} \) - \( \frac{5 x 3}{15 x 3} \)

\( \frac{25}{45} \) - \( \frac{15}{45} \)

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

\( \frac{25 - 15}{45} \) = \( \frac{10}{45} \) = \(\frac{2}{9}\)

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

vans = \( \frac{38}{9} \) = 4\(\frac{2}{9}\)

So, it will take 4 full vans and one partially full van to transport the entire team making a total of 5 vans.

1

The area of a rectangle is width (w) x height (h). In this problem we know that the rectangle is twice as long as it is wide so h = 2w. The perimeter of a rectangle is 2w + 2h and we know that the perimeter of this rectangle is 12 meters so the equation becomes: 2w + 2h = 12.

Putting these two equations together and solving for width (w):

2w + 2h = 12
w + h = \( \frac{12}{2} \)
w + h = 6
w = 6 - h

From the question we know that h = 2w so substituting 2w for h gives us:

w = 6 - 2w
3w = 6
w = \( \frac{6}{3} \)
w = 2

Since h = 2w that makes h = (2 x 2) = 4 and the area = h x w = 2 x 4 = 8 m²